#!/usr/bin/env python
# Copyright 2014-2019 The PySCF Developers. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
# Author: Qiming Sun <osirpt.sun@gmail.com>
#
'''
Non-relativistic RKS analytical Hessian
'''
import numpy
import ctypes
from pyscf import lib
from pyscf import gto
from pyscf.lib import logger
from pyscf.hessian import rhf as rhf_hess
from pyscf.grad import rks as rks_grad
from pyscf.dft import numint, gen_grid
# import pyscf.grad.rks to activate nuc_grad_method method
from pyscf.grad import rks # noqa
min_grid_blksize = 128*128
NLC_REMOVE_ZERO_RHO_GRID_THRESHOLD = 1e-8
libdft = lib.load_library('libdft')
contract = numpy.einsum
[docs]
def partial_hess_elec(hessobj, mo_energy=None, mo_coeff=None, mo_occ=None,
atmlst=None, max_memory=4000, verbose=None):
log = logger.new_logger(hessobj, verbose)
time0 = t1 = (logger.process_clock(), logger.perf_counter())
mol = hessobj.mol
mf = hessobj.base
ni = mf._numint
if mo_energy is None: mo_energy = mf.mo_energy
if mo_occ is None: mo_occ = mf.mo_occ
if mo_coeff is None: mo_coeff = mf.mo_coeff
if atmlst is None: atmlst = range(mol.natm)
nao, nmo = mo_coeff.shape
mocc = mo_coeff[:,mo_occ>0]
dm0 = numpy.dot(mocc, mocc.T) * 2
omega, alpha, hyb = ni.rsh_and_hybrid_coeff(mf.xc, spin=mol.spin)
hybrid = ni.libxc.is_hybrid_xc(mf.xc)
de2, ej, ek = rhf_hess._partial_hess_ejk(hessobj, mo_energy, mo_coeff, mo_occ,
atmlst, max_memory, verbose,
with_k=hybrid)
de2 += ej - hyb * ek # (A,B,dR_A,dR_B)
mem_now = lib.current_memory()[0]
max_memory = max(2000, mf.max_memory*.9-mem_now)
veff_diag = _get_vxc_diag(hessobj, mo_coeff, mo_occ, max_memory)
if hybrid and omega != 0:
with mol.with_range_coulomb(omega):
vk1 = rhf_hess._get_jk(mol, 'int2e_ipip1', 9, 's2kl',
['jk->s1il', dm0])[0]
veff_diag -= (alpha-hyb)*.5 * vk1.reshape(3,3,nao,nao)
vk1 = None
t1 = log.timer_debug1('contracting int2e_ipip1', *t1)
aoslices = mol.aoslice_by_atom()
vxc = _get_vxc_deriv2(hessobj, mo_coeff, mo_occ, max_memory)
for i0, ia in enumerate(atmlst):
shl0, shl1, p0, p1 = aoslices[ia]
shls_slice = (shl0, shl1) + (0, mol.nbas)*3
veff = vxc[ia]
if hybrid and omega != 0:
with mol.with_range_coulomb(omega):
vk1, vk2 = rhf_hess._get_jk(mol, 'int2e_ip1ip2', 9, 's1',
['li->s1kj', dm0[:,p0:p1], # vk1
'lj->s1ki', dm0 ], # vk2
shls_slice=shls_slice)
veff -= (alpha-hyb)*.5 * vk1.reshape(3,3,nao,nao)
veff[:,:,:,p0:p1] -= (alpha-hyb)*.5 * vk2.reshape(3,3,nao,p1-p0)
t1 = log.timer_debug1('range-separated int2e_ip1ip2 for atom %d'%ia, *t1)
with mol.with_range_coulomb(omega):
vk1 = rhf_hess._get_jk(mol, 'int2e_ipvip1', 9, 's2kl',
['li->s1kj', dm0[:,p0:p1]], # vk1
shls_slice=shls_slice)[0]
veff -= (alpha-hyb)*.5 * vk1.transpose(0,2,1).reshape(3,3,nao,nao)
t1 = log.timer_debug1('range-separated int2e_ipvip1 for atom %d'%ia, *t1)
vk1 = vk2 = None
de2[i0,i0] += numpy.einsum('xypq,pq->xy', veff_diag[:,:,p0:p1], dm0[p0:p1])*2
for j0, ja in enumerate(atmlst[:i0+1]):
q0, q1 = aoslices[ja][2:]
de2[i0,j0] += numpy.einsum('xypq,pq->xy', veff[:,:,q0:q1], dm0[q0:q1])*2
for j0 in range(i0):
de2[j0,i0] = de2[i0,j0].T
if mf.do_nlc():
de2 += _get_enlc_deriv2(hessobj, mo_coeff, mo_occ, max_memory)
log.timer('RKS partial hessian', *time0)
return de2
[docs]
def make_h1(hessobj, mo_coeff, mo_occ, chkfile=None, atmlst=None, verbose=None):
mol = hessobj.mol
if atmlst is None:
atmlst = range(mol.natm)
nao, nmo = mo_coeff.shape
mocc = mo_coeff[:,mo_occ>0]
dm0 = numpy.dot(mocc, mocc.T) * 2
hcore_deriv = hessobj.base.nuc_grad_method().hcore_generator(mol)
mf = hessobj.base
ni = mf._numint
ni.libxc.test_deriv_order(mf.xc, 2, raise_error=True)
omega, alpha, hyb = ni.rsh_and_hybrid_coeff(mf.xc, spin=mol.spin)
hybrid = ni.libxc.is_hybrid_xc(mf.xc)
mem_now = lib.current_memory()[0]
max_memory = max(2000, mf.max_memory*.9-mem_now)
h1ao = _get_vxc_deriv1(hessobj, mo_coeff, mo_occ, max_memory)
if mf.do_nlc():
h1ao += _get_vnlc_deriv1(hessobj, mo_coeff, mo_occ, max_memory)
aoslices = mol.aoslice_by_atom()
for i0, ia in enumerate(atmlst):
shl0, shl1, p0, p1 = aoslices[ia]
shls_slice = (shl0, shl1) + (0, mol.nbas)*3
if hybrid:
vj1, vj2, vk1, vk2 = \
rhf_hess._get_jk(mol, 'int2e_ip1', 3, 's2kl',
['ji->s2kl', -dm0[:,p0:p1], # vj1
'lk->s1ij', -dm0 , # vj2
'li->s1kj', -dm0[:,p0:p1], # vk1
'jk->s1il', -dm0 ], # vk2
shls_slice=shls_slice)
veff = vj1 - hyb * .5 * vk1
veff[:,p0:p1] += vj2 - hyb * .5 * vk2
if omega != 0:
with mol.with_range_coulomb(omega):
vk1, vk2 = \
rhf_hess._get_jk(mol, 'int2e_ip1', 3, 's2kl',
['li->s1kj', -dm0[:,p0:p1], # vk1
'jk->s1il', -dm0 ], # vk2
shls_slice=shls_slice)
veff -= (alpha-hyb) * .5 * vk1
veff[:,p0:p1] -= (alpha-hyb) * .5 * vk2
else:
vj1, vj2 = rhf_hess._get_jk(mol, 'int2e_ip1', 3, 's2kl',
['ji->s2kl', -dm0[:,p0:p1], # vj1
'lk->s1ij', -dm0 ], # vj2
shls_slice=shls_slice)
veff = vj1
veff[:,p0:p1] += vj2
h1ao[ia] += veff + veff.transpose(0,2,1)
h1ao[ia] += hcore_deriv(ia)
return h1ao
XX, XY, XZ = 4, 5, 6
YX, YY, YZ = 5, 7, 8
ZX, ZY, ZZ = 6, 8, 9
XXX, XXY, XXZ, XYY, XYZ, XZZ = 10, 11, 12, 13, 14, 15
YYY, YYZ, YZZ, ZZZ = 16, 17, 18, 19
def _get_vxc_diag(hessobj, mo_coeff, mo_occ, max_memory):
mol = hessobj.mol
mf = hessobj.base
if hessobj.grids is not None:
grids = hessobj.grids
else:
grids = mf.grids
if grids.coords is None:
grids.build(with_non0tab=True)
nao, nmo = mo_coeff.shape
ni = mf._numint
xctype = ni._xc_type(mf.xc)
shls_slice = (0, mol.nbas)
ao_loc = mol.ao_loc_nr()
vmat = numpy.zeros((6,nao,nao))
if xctype == 'LDA':
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[0], mo_coeff, mo_occ, mask, xctype)
vxc = ni.eval_xc_eff(mf.xc, rho, 1, xctype=xctype)[1]
wv = weight * vxc[0]
aow = numint._scale_ao(ao[0], wv)
for i in range(6):
vmat[i] += numint._dot_ao_ao(mol, ao[i+4], aow, mask, shls_slice, ao_loc)
aow = None
elif xctype == 'GGA':
def contract_(mat, ao, aoidx, wv, mask):
aow = numint._scale_ao(ao[aoidx[0]], wv[1])
aow+= numint._scale_ao(ao[aoidx[1]], wv[2])
aow+= numint._scale_ao(ao[aoidx[2]], wv[3])
mat += numint._dot_ao_ao(mol, aow, ao[0], mask, shls_slice, ao_loc)
ao_deriv = 3
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:4], mo_coeff, mo_occ, mask, xctype)
vxc = ni.eval_xc_eff(mf.xc, rho, 1, xctype=xctype)[1]
wv = weight * vxc
#:aow = numpy.einsum('npi,np->pi', ao[:4], wv[:4])
aow = numint._scale_ao(ao[:4], wv[:4])
for i in range(6):
vmat[i] += numint._dot_ao_ao(mol, ao[i+4], aow, mask, shls_slice, ao_loc)
contract_(vmat[0], ao, [XXX,XXY,XXZ], wv, mask)
contract_(vmat[1], ao, [XXY,XYY,XYZ], wv, mask)
contract_(vmat[2], ao, [XXZ,XYZ,XZZ], wv, mask)
contract_(vmat[3], ao, [XYY,YYY,YYZ], wv, mask)
contract_(vmat[4], ao, [XYZ,YYZ,YZZ], wv, mask)
contract_(vmat[5], ao, [XZZ,YZZ,ZZZ], wv, mask)
rho = vxc = wv = aow = None
elif xctype == 'MGGA':
def contract_(mat, ao, aoidx, wv, mask):
aow = numint._scale_ao(ao[aoidx[0]], wv[1])
aow+= numint._scale_ao(ao[aoidx[1]], wv[2])
aow+= numint._scale_ao(ao[aoidx[2]], wv[3])
mat += numint._dot_ao_ao(mol, aow, ao[0], mask, shls_slice, ao_loc)
ao_deriv = 3
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:10], mo_coeff, mo_occ, mask, xctype)
vxc = ni.eval_xc_eff(mf.xc, rho, 1, xctype=xctype)[1]
wv = weight * vxc
wv[4] *= .5 # for the factor 1/2 in tau
#:aow = numpy.einsum('npi,np->pi', ao[:4], wv[:4])
aow = numint._scale_ao(ao[:4], wv[:4])
for i in range(6):
vmat[i] += numint._dot_ao_ao(mol, ao[i+4], aow, mask, shls_slice, ao_loc)
contract_(vmat[0], ao, [XXX,XXY,XXZ], wv, mask)
contract_(vmat[1], ao, [XXY,XYY,XYZ], wv, mask)
contract_(vmat[2], ao, [XXZ,XYZ,XZZ], wv, mask)
contract_(vmat[3], ao, [XYY,YYY,YYZ], wv, mask)
contract_(vmat[4], ao, [XYZ,YYZ,YZZ], wv, mask)
contract_(vmat[5], ao, [XZZ,YZZ,ZZZ], wv, mask)
aow = [numint._scale_ao(ao[i], wv[4]) for i in range(1, 4)]
for i, j in enumerate([XXX, XXY, XXZ, XYY, XYZ, XZZ]):
vmat[i] += numint._dot_ao_ao(mol, ao[j], aow[0], mask, shls_slice, ao_loc)
for i, j in enumerate([XXY, XYY, XYZ, YYY, YYZ, YZZ]):
vmat[i] += numint._dot_ao_ao(mol, ao[j], aow[1], mask, shls_slice, ao_loc)
for i, j in enumerate([XXZ, XYZ, XZZ, YYZ, YZZ, ZZZ]):
vmat[i] += numint._dot_ao_ao(mol, ao[j], aow[2], mask, shls_slice, ao_loc)
vmat = vmat[[0,1,2,
1,3,4,
2,4,5]]
return vmat.reshape(3,3,nao,nao)
def _make_dR_rho1(ao, ao_dm0, atm_id, aoslices, xctype):
p0, p1 = aoslices[atm_id][2:]
ngrids = ao[0].shape[0]
if xctype == 'GGA':
rho1 = numpy.zeros((3,4,ngrids))
elif xctype == 'MGGA':
rho1 = numpy.zeros((3,5,ngrids))
ao_dm0_x = ao_dm0[1][:,p0:p1]
ao_dm0_y = ao_dm0[2][:,p0:p1]
ao_dm0_z = ao_dm0[3][:,p0:p1]
# (d_X \nabla mu) dot \nalba nu DM_{mu,nu}
rho1[0,4] += numpy.einsum('pi,pi->p', ao[XX,:,p0:p1], ao_dm0_x)
rho1[0,4] += numpy.einsum('pi,pi->p', ao[XY,:,p0:p1], ao_dm0_y)
rho1[0,4] += numpy.einsum('pi,pi->p', ao[XZ,:,p0:p1], ao_dm0_z)
rho1[1,4] += numpy.einsum('pi,pi->p', ao[YX,:,p0:p1], ao_dm0_x)
rho1[1,4] += numpy.einsum('pi,pi->p', ao[YY,:,p0:p1], ao_dm0_y)
rho1[1,4] += numpy.einsum('pi,pi->p', ao[YZ,:,p0:p1], ao_dm0_z)
rho1[2,4] += numpy.einsum('pi,pi->p', ao[ZX,:,p0:p1], ao_dm0_x)
rho1[2,4] += numpy.einsum('pi,pi->p', ao[ZY,:,p0:p1], ao_dm0_y)
rho1[2,4] += numpy.einsum('pi,pi->p', ao[ZZ,:,p0:p1], ao_dm0_z)
rho1[:,4] *= .5
else:
raise RuntimeError
ao_dm0_0 = ao_dm0[0][:,p0:p1]
# (d_X \nabla_x mu) nu DM_{mu,nu}
rho1[:,0] = numpy.einsum('xpi,pi->xp', ao[1:4,:,p0:p1], ao_dm0_0)
rho1[0,1]+= numpy.einsum('pi,pi->p', ao[XX,:,p0:p1], ao_dm0_0)
rho1[0,2]+= numpy.einsum('pi,pi->p', ao[XY,:,p0:p1], ao_dm0_0)
rho1[0,3]+= numpy.einsum('pi,pi->p', ao[XZ,:,p0:p1], ao_dm0_0)
rho1[1,1]+= numpy.einsum('pi,pi->p', ao[YX,:,p0:p1], ao_dm0_0)
rho1[1,2]+= numpy.einsum('pi,pi->p', ao[YY,:,p0:p1], ao_dm0_0)
rho1[1,3]+= numpy.einsum('pi,pi->p', ao[YZ,:,p0:p1], ao_dm0_0)
rho1[2,1]+= numpy.einsum('pi,pi->p', ao[ZX,:,p0:p1], ao_dm0_0)
rho1[2,2]+= numpy.einsum('pi,pi->p', ao[ZY,:,p0:p1], ao_dm0_0)
rho1[2,3]+= numpy.einsum('pi,pi->p', ao[ZZ,:,p0:p1], ao_dm0_0)
# (d_X mu) (\nabla_x nu) DM_{mu,nu}
rho1[:,1] += numpy.einsum('xpi,pi->xp', ao[1:4,:,p0:p1], ao_dm0[1][:,p0:p1])
rho1[:,2] += numpy.einsum('xpi,pi->xp', ao[1:4,:,p0:p1], ao_dm0[2][:,p0:p1])
rho1[:,3] += numpy.einsum('xpi,pi->xp', ao[1:4,:,p0:p1], ao_dm0[3][:,p0:p1])
# *2 for |mu> DM <d_X nu|
return rho1 * 2
def _d1d2_dot_(vmat, mol, ao1, ao2, mask, ao_loc, dR1_on_bra=True):
shls_slice = (0, mol.nbas)
if dR1_on_bra: # (d/dR1 bra) * (d/dR2 ket)
for d1 in range(3):
for d2 in range(3):
vmat[d1,d2] += numint._dot_ao_ao(mol, ao1[d1], ao2[d2], mask,
shls_slice, ao_loc)
else: # (d/dR2 bra) * (d/dR1 ket)
for d1 in range(3):
for d2 in range(3):
vmat[d1,d2] += numint._dot_ao_ao(mol, ao1[d2], ao2[d1], mask,
shls_slice, ao_loc)
def _get_vxc_deriv2(hessobj, mo_coeff, mo_occ, max_memory):
mol = hessobj.mol
mf = hessobj.base
if hessobj.grids is not None:
grids = hessobj.grids
else:
grids = mf.grids
if grids.coords is None:
grids.build(with_non0tab=True)
nao, nmo = mo_coeff.shape
ni = mf._numint
xctype = ni._xc_type(mf.xc)
aoslices = mol.aoslice_by_atom()
shls_slice = (0, mol.nbas)
ao_loc = mol.ao_loc_nr()
dm0 = mf.make_rdm1(mo_coeff, mo_occ)
vmat = numpy.zeros((mol.natm,3,3,nao,nao))
ipip = numpy.zeros((3,3,nao,nao))
if xctype == 'LDA':
ao_deriv = 1
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[0], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc_eff(mf.xc, rho, 2, xctype=xctype)[1:3]
wv = weight * vxc[0]
aow = [numint._scale_ao(ao[i], wv) for i in range(1, 4)]
_d1d2_dot_(ipip, mol, aow, ao[1:4], mask, ao_loc, False)
ao_dm0 = numint._dot_ao_dm(mol, ao[0], dm0, mask, shls_slice, ao_loc)
wf = weight * fxc[0,0]
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
# *2 for \nabla|ket> in rho1
rho1 = numpy.einsum('xpi,pi->xp', ao[1:,:,p0:p1], ao_dm0[:,p0:p1]) * 2
# aow ~ rho1 ~ d/dR1
wv = wf * rho1
aow = [numint._scale_ao(ao[0], wv[i]) for i in range(3)]
_d1d2_dot_(vmat[ia], mol, ao[1:4], aow, mask, ao_loc, False)
ao_dm0 = aow = None
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmat[ia,:,:,:,p0:p1] += ipip[:,:,:,p0:p1]
elif xctype == 'GGA':
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:4], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc_eff(mf.xc, rho, 2, xctype=xctype)[1:3]
wv = weight * vxc
wv[0] *= .5
aow = rks_grad._make_dR_dao_w(ao, wv)
_d1d2_dot_(ipip, mol, aow, ao[1:4], mask, ao_loc, False)
ao_dm0 = [numint._dot_ao_dm(mol, ao[i], dm0, mask, shls_slice, ao_loc) for i in range(4)]
wf = weight * fxc
for ia in range(mol.natm):
dR_rho1 = _make_dR_rho1(ao, ao_dm0, ia, aoslices, xctype)
wv = numpy.einsum('xyg,sxg->syg', wf, dR_rho1)
wv[:,0] *= .5
for i in range(3):
aow = rks_grad._make_dR_dao_w(ao, wv[i])
rks_grad._d1_dot_(vmat[ia,i], mol, aow, ao[0], mask, ao_loc, True)
aow = [numint._scale_ao(ao[:4], wv[i,:4]) for i in range(3)]
_d1d2_dot_(vmat[ia], mol, ao[1:4], aow, mask, ao_loc, False)
ao_dm0 = aow = None
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmat[ia,:,:,:,p0:p1] += ipip[:,:,:,p0:p1]
vmat[ia,:,:,:,p0:p1] += ipip[:,:,p0:p1].transpose(1,0,3,2)
elif xctype == 'MGGA':
XX, XY, XZ = 4, 5, 6
YX, YY, YZ = 5, 7, 8
ZX, ZY, ZZ = 6, 8, 9
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:10], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc_eff(mf.xc, rho, 2, xctype=xctype)[1:3]
wv = weight * vxc
wv[0] *= .5
wv[4] *= .25
aow = rks_grad._make_dR_dao_w(ao, wv)
_d1d2_dot_(ipip, mol, aow, ao[1:4], mask, ao_loc, False)
aow = [numint._scale_ao(ao[i], wv[4]) for i in range(4, 10)]
_d1d2_dot_(ipip, mol, [aow[0], aow[1], aow[2]], [ao[XX], ao[XY], ao[XZ]], mask, ao_loc, False)
_d1d2_dot_(ipip, mol, [aow[1], aow[3], aow[4]], [ao[YX], ao[YY], ao[YZ]], mask, ao_loc, False)
_d1d2_dot_(ipip, mol, [aow[2], aow[4], aow[5]], [ao[ZX], ao[ZY], ao[ZZ]], mask, ao_loc, False)
ao_dm0 = [numint._dot_ao_dm(mol, ao[i], dm0, mask, shls_slice, ao_loc) for i in range(4)]
wf = weight * fxc
for ia in range(mol.natm):
dR_rho1 = _make_dR_rho1(ao, ao_dm0, ia, aoslices, xctype)
wv = numpy.einsum('xyg,sxg->syg', wf, dR_rho1)
wv[:,0] *= .5
wv[:,4] *= .5 # for the factor 1/2 in tau
for i in range(3):
aow = rks_grad._make_dR_dao_w(ao, wv[i])
rks_grad._d1_dot_(vmat[ia,i], mol, aow, ao[0], mask, ao_loc, True)
aow = [numint._scale_ao(ao[:4], wv[i,:4]) for i in range(3)]
_d1d2_dot_(vmat[ia], mol, ao[1:4], aow, mask, ao_loc, False)
aow = [numint._scale_ao(ao[1], wv[i,4]) for i in range(3)]
_d1d2_dot_(vmat[ia], mol, [ao[XX], ao[XY], ao[XZ]], aow, mask, ao_loc, False)
aow = [numint._scale_ao(ao[2], wv[i,4]) for i in range(3)]
_d1d2_dot_(vmat[ia], mol, [ao[YX], ao[YY], ao[YZ]], aow, mask, ao_loc, False)
aow = [numint._scale_ao(ao[3], wv[i,4]) for i in range(3)]
_d1d2_dot_(vmat[ia], mol, [ao[ZX], ao[ZY], ao[ZZ]], aow, mask, ao_loc, False)
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmat[ia,:,:,:,p0:p1] += ipip[:,:,:,p0:p1]
vmat[ia,:,:,:,p0:p1] += ipip[:,:,p0:p1].transpose(1,0,3,2)
return vmat
def _get_enlc_deriv2_numerical(hessobj, mo_coeff, mo_occ, max_memory):
"""
Attention: Numerical nlc energy 2nd derivative includes grid response.
"""
mol = hessobj.mol
mf = hessobj.base
mocc = mo_coeff[:,mo_occ>0]
dm0 = numpy.dot(mocc, mocc.T) * 2
de2 = numpy.empty([mol.natm, mol.natm, 3, 3])
def get_nlc_de(grad_obj, dm):
from pyscf.grad.rks import _initialize_grids, get_nlc_vxc_full_response
mol = grad_obj.mol
mf = grad_obj.base
ni = mf._numint
_, nlcgrids = _initialize_grids(grad_obj)
if ni.libxc.is_nlc(mf.xc):
xc = mf.xc
else:
xc = mf.nlc
enlc, vnlc = get_nlc_vxc_full_response(
ni, mol, nlcgrids, xc, dm,
max_memory=max_memory, verbose=grad_obj.verbose)
aoslices = mol.aoslice_by_atom()
de = numpy.zeros((mol.natm,3))
for i_atom in range(mol.natm):
p0, p1 = aoslices[i_atom, 2:]
de[i_atom] += numpy.einsum('xij,ij->x', vnlc[:,p0:p1], dm[p0:p1]) * 2
assert enlc is not None
de += enlc
return de
dx = 1e-3
mol_copy = mol.copy()
mol_copy.verbose = 0
grad_obj = mf.Gradients()
for i_atom in range(mol.natm):
for i_xyz in range(3):
xyz_p = mol.atom_coords()
xyz_p[i_atom, i_xyz] += dx
mol_copy.set_geom_(xyz_p, unit='Bohr')
grad_obj.reset(mol_copy)
de_p = get_nlc_de(grad_obj, dm0)
xyz_m = mol.atom_coords()
xyz_m[i_atom, i_xyz] -= dx
mol_copy.set_geom_(xyz_m, unit='Bohr')
mol_copy.build()
grad_obj.reset(mol_copy)
de_m = get_nlc_de(grad_obj, dm0)
de2[i_atom, :, i_xyz, :] = (de_p - de_m) / (2 * dx)
grad_obj.reset(mol)
return de2
[docs]
def get_d2mu_dr2(ao):
assert ao.ndim == 3
nao = ao.shape[1]
ngrids = ao.shape[2]
d2mu_dr2 = numpy.empty([3, 3, nao, ngrids])
d2mu_dr2[0,0,:,:] = ao[XX, :, :]
d2mu_dr2[0,1,:,:] = ao[XY, :, :]
d2mu_dr2[1,0,:,:] = ao[XY, :, :]
d2mu_dr2[0,2,:,:] = ao[XZ, :, :]
d2mu_dr2[2,0,:,:] = ao[XZ, :, :]
d2mu_dr2[1,1,:,:] = ao[YY, :, :]
d2mu_dr2[1,2,:,:] = ao[YZ, :, :]
d2mu_dr2[2,1,:,:] = ao[YZ, :, :]
d2mu_dr2[2,2,:,:] = ao[ZZ, :, :]
return d2mu_dr2
[docs]
def get_d3mu_dr3(ao):
assert ao.ndim == 3
nao = ao.shape[1]
ngrids = ao.shape[2]
d3mu_dr3 = numpy.empty([3, 3, 3, nao, ngrids])
d3mu_dr3[0,0,0,:,:] = ao[XXX,:,:]
d3mu_dr3[0,0,1,:,:] = ao[XXY,:,:]
d3mu_dr3[0,1,0,:,:] = ao[XXY,:,:]
d3mu_dr3[1,0,0,:,:] = ao[XXY,:,:]
d3mu_dr3[0,0,2,:,:] = ao[XXZ,:,:]
d3mu_dr3[0,2,0,:,:] = ao[XXZ,:,:]
d3mu_dr3[2,0,0,:,:] = ao[XXZ,:,:]
d3mu_dr3[0,1,1,:,:] = ao[XYY,:,:]
d3mu_dr3[1,0,1,:,:] = ao[XYY,:,:]
d3mu_dr3[1,1,0,:,:] = ao[XYY,:,:]
d3mu_dr3[0,1,2,:,:] = ao[XYZ,:,:]
d3mu_dr3[1,0,2,:,:] = ao[XYZ,:,:]
d3mu_dr3[1,2,0,:,:] = ao[XYZ,:,:]
d3mu_dr3[0,2,1,:,:] = ao[XYZ,:,:]
d3mu_dr3[2,0,1,:,:] = ao[XYZ,:,:]
d3mu_dr3[2,1,0,:,:] = ao[XYZ,:,:]
d3mu_dr3[0,2,2,:,:] = ao[XZZ,:,:]
d3mu_dr3[2,0,2,:,:] = ao[XZZ,:,:]
d3mu_dr3[2,2,0,:,:] = ao[XZZ,:,:]
d3mu_dr3[1,1,1,:,:] = ao[YYY,:,:]
d3mu_dr3[1,1,2,:,:] = ao[YYZ,:,:]
d3mu_dr3[1,2,1,:,:] = ao[YYZ,:,:]
d3mu_dr3[2,1,1,:,:] = ao[YYZ,:,:]
d3mu_dr3[1,2,2,:,:] = ao[YZZ,:,:]
d3mu_dr3[2,1,2,:,:] = ao[YZZ,:,:]
d3mu_dr3[2,2,1,:,:] = ao[YZZ,:,:]
d3mu_dr3[2,2,2,:,:] = ao[ZZZ,:,:]
return d3mu_dr3
[docs]
def get_d2rho_dAdr_orbital_response(d2mu_dr2, dmu_dr, mu, dm0, aoslices):
assert mu.ndim == 2
nao = mu.shape[0]
ngrids = mu.shape[1]
natm = len(aoslices)
assert d2mu_dr2.shape == (3, 3, nao, ngrids)
assert dmu_dr.shape == (3, nao, ngrids)
assert dm0.shape == (nao, nao)
d2rho_dAdr = numpy.zeros([natm, 3, 3, ngrids])
for i_atom in range(natm):
p0, p1 = aoslices[i_atom][2:]
# d2rho_dAdr[i_atom, :, :, :] += numpy.einsum('dDig,jg,ij->dDg', -d2mu_dr2[:, :, p0:p1, :], mu, dm0[p0:p1, :])
# d2rho_dAdr[i_atom, :, :, :] += numpy.einsum('dDig,jg,ij->dDg', -d2mu_dr2[:, :, p0:p1, :], mu, dm0[:, p0:p1].T)
# d2rho_dAdr[i_atom, :, :, :] += numpy.einsum('dig,Djg,ij->dDg', -dmu_dr[:, p0:p1, :], dmu_dr, dm0[p0:p1, :])
# d2rho_dAdr[i_atom, :, :, :] += numpy.einsum('dig,Djg,ij->dDg', -dmu_dr[:, p0:p1, :], dmu_dr, dm0[:, p0:p1].T)
nu_dot_dm = dm0[p0:p1, :] @ mu
d2rho_dAdr[i_atom, :, :, :] += contract('dDig,ig->dDg', -d2mu_dr2[:, :, p0:p1, :], nu_dot_dm)
nu_dot_dm = None
mu_dot_dm = dm0[:, p0:p1].T @ mu
d2rho_dAdr[i_atom, :, :, :] += contract('dDig,ig->dDg', -d2mu_dr2[:, :, p0:p1, :], mu_dot_dm)
mu_dot_dm = None
dnudr_dot_dm = contract('djg,ij->dig', dmu_dr, dm0[p0:p1, :])
d2rho_dAdr[i_atom, :, :, :] += contract('dig,Dig->dDg', -dmu_dr[:, p0:p1, :], dnudr_dot_dm)
dnudr_dot_dm = None
dmudr_dot_dm = contract('djg,ij->dig', dmu_dr, dm0[:, p0:p1].T)
d2rho_dAdr[i_atom, :, :, :] += contract('dig,Dig->dDg', -dmu_dr[:, p0:p1, :], dmudr_dot_dm)
dmudr_dot_dm = None
return d2rho_dAdr
[docs]
def get_d2rho_dAdr_grid_response(d2mu_dr2, dmu_dr, mu, dm0, atom_to_grid_index_map = None, i_atom = None):
assert mu.ndim == 2
nao = mu.shape[0]
ngrids = mu.shape[1]
assert d2mu_dr2.shape == (3, 3, nao, ngrids)
assert dmu_dr.shape == (3, nao, ngrids)
assert dm0.shape == (nao, nao)
if i_atom is None:
assert atom_to_grid_index_map is not None
natm = len(atom_to_grid_index_map)
d2rho_dAdr_grid_response = numpy.zeros([natm, 3, 3, ngrids])
for i_atom in range(natm):
associated_grid_index = atom_to_grid_index_map[i_atom]
# d2rho_dAdr_response = numpy.einsum('dDig,jg,ij->dDg',
# d2mu_dr2[:, :, :, associated_grid_index], mu[:, associated_grid_index], dm0)
# d2rho_dAdr_response += numpy.einsum('dDig,jg,ij->dDg',
# d2mu_dr2[:, :, :, associated_grid_index], mu[:, associated_grid_index], dm0.T)
# d2rho_dAdr_response += numpy.einsum('dig,Djg,ij->dDg',
# dmu_dr[:, :, associated_grid_index], dmu_dr[:, :, associated_grid_index], dm0)
# d2rho_dAdr_response += numpy.einsum('dig,Djg,ij->dDg',
# dmu_dr[:, :, associated_grid_index], dmu_dr[:, :, associated_grid_index], dm0.T)
dm_dot_mu_and_nu = (dm0 + dm0.T) @ mu[:, associated_grid_index]
d2rho_dAdr_response = contract('dDig,ig->dDg', d2mu_dr2[:, :, :, associated_grid_index], dm_dot_mu_and_nu)
dm_dot_mu_and_nu = None
dm_dot_dmu_and_dnu = contract('djg,ij->dig', dmu_dr[:, :, associated_grid_index], dm0 + dm0.T)
d2rho_dAdr_response += contract('dig,Dig->dDg', dmu_dr[:, :, associated_grid_index], dm_dot_dmu_and_dnu)
dm_dot_dmu_and_dnu = None
d2rho_dAdr_grid_response[i_atom][:, :, associated_grid_index] = d2rho_dAdr_response
else:
assert atom_to_grid_index_map is None
# Here we assume all grids belong to atom i
dm_dot_mu_and_nu = (dm0 + dm0.T) @ mu
d2rho_dAdr_grid_response = contract('dDig,ig->dDg', d2mu_dr2, dm_dot_mu_and_nu)
dm_dot_mu_and_nu = None
dm_dot_dmu_and_dnu = contract('djg,ij->dig', dmu_dr, dm0 + dm0.T)
d2rho_dAdr_grid_response += contract('dig,Dig->dDg', dmu_dr, dm_dot_dmu_and_dnu)
dm_dot_dmu_and_dnu = None
return d2rho_dAdr_grid_response
[docs]
def get_drhodA_dgammadA_orbital_response(d2mu_dr2, dmu_dr, mu, drho_dr, dm0, aoslices):
assert mu.ndim == 2
nao = mu.shape[0]
ngrids = mu.shape[1]
natm = len(aoslices)
assert d2mu_dr2.shape == (3, 3, nao, ngrids)
assert dmu_dr.shape == (3, nao, ngrids)
assert drho_dr.shape == (3, ngrids)
assert dm0.shape == (nao, nao)
drhodr_dot_dmudr = contract('Djg,Dg->jg', dmu_dr, drho_dr)
drho_dA = numpy.zeros([natm, 3, ngrids])
dgamma_dA = numpy.zeros([natm, 3, ngrids])
for i_atom in range(natm):
p0, p1 = aoslices[i_atom][2:]
# drho_dA[i_atom, :, :] += numpy.einsum('dig,jg,ij->dg', -dmu_dr[:, p0:p1, :], mu, dm0[p0:p1, :])
# drho_dA[i_atom, :, :] += numpy.einsum('dig,jg,ij->dg', -dmu_dr[:, p0:p1, :], mu, dm0[:, p0:p1].T)
nu_dot_dm = dm0[p0:p1, :] @ mu
drho_dA[i_atom, :, :] += contract('dig,ig->dg', -dmu_dr[:, p0:p1, :], nu_dot_dm)
mu_dot_dm = dm0[:, p0:p1].T @ mu
drho_dA[i_atom, :, :] += contract('dig,ig->dg', -dmu_dr[:, p0:p1, :], mu_dot_dm)
# dgamma_dA[i_atom, :, :] += numpy.einsum('dDig,jg,Dg,ij->dg',
# -d2mu_dr2[:, :, p0:p1, :], mu, drho_dr, dm0[p0:p1, :])
# dgamma_dA[i_atom, :, :] += numpy.einsum('dDig,jg,Dg,ij->dg',
# -d2mu_dr2[:, :, p0:p1, :], mu, drho_dr, dm0[:, p0:p1].T)
# dgamma_dA[i_atom, :, :] += numpy.einsum('dig,Djg,Dg,ij->dg',
# -dmu_dr[:, p0:p1, :], dmu_dr, drho_dr, dm0[p0:p1, :])
# dgamma_dA[i_atom, :, :] += numpy.einsum('dig,Djg,Dg,ij->dg',
# -dmu_dr[:, p0:p1, :], dmu_dr, drho_dr, dm0[:, p0:p1].T)
d2mudAdr_dot_drhodr = contract('dDig,Dg->dig', -d2mu_dr2[:, :, p0:p1, :], drho_dr)
dgamma_dA[i_atom, :, :] += contract('dig,ig->dg', d2mudAdr_dot_drhodr, nu_dot_dm)
dgamma_dA[i_atom, :, :] += contract('dig,ig->dg', d2mudAdr_dot_drhodr, mu_dot_dm)
d2mudAdr_dot_drhodr = None
nu_dot_dm = None
mu_dot_dm = None
drhodr_dot_dnudr_dot_dm = dm0[p0:p1, :] @ drhodr_dot_dmudr
dgamma_dA[i_atom, :, :] += contract('dig,ig->dg', -dmu_dr[:, p0:p1, :], drhodr_dot_dnudr_dot_dm)
drhodr_dot_dnudr_dot_dm = None
drhodr_dot_dmudr_dot_dm = dm0[:, p0:p1].T @ drhodr_dot_dmudr
dgamma_dA[i_atom, :, :] += contract('dig,ig->dg', -dmu_dr[:, p0:p1, :], drhodr_dot_dmudr_dot_dm)
drhodr_dot_dmudr_dot_dm = None
dgamma_dA *= 2
return drho_dA, dgamma_dA
[docs]
def get_drhodA_dgammadA_grid_response(d2mu_dr2, dmu_dr, mu, drho_dr, dm0, atom_to_grid_index_map = None, i_atom = None):
assert mu.ndim == 2
nao = mu.shape[0]
ngrids = mu.shape[1]
assert d2mu_dr2.shape == (3, 3, nao, ngrids)
assert dmu_dr.shape == (3, nao, ngrids)
assert drho_dr.shape == (3, ngrids)
assert dm0.shape == (nao, nao)
if i_atom is None:
assert atom_to_grid_index_map is not None
natm = len(atom_to_grid_index_map)
drho_dA_grid_response = numpy.zeros([natm, 3, ngrids])
dgamma_dA_grid_response = numpy.zeros([natm, 3, ngrids])
for i_atom in range(natm):
associated_grid_index = atom_to_grid_index_map[i_atom]
# rho_response = numpy.einsum('dig,jg,ij->dg',
# dmu_dr[:, :, associated_grid_index], mu[:, associated_grid_index], dm0)
# rho_response += numpy.einsum('dig,jg,ij->dg',
# dmu_dr[:, :, associated_grid_index], mu[:, associated_grid_index], dm0.T)
dm_dot_mu_and_nu = (dm0 + dm0.T) @ mu[:, associated_grid_index]
rho_response = contract('dig,ig->dg', dmu_dr[:, :, associated_grid_index], dm_dot_mu_and_nu)
drho_dA_grid_response[i_atom][:, associated_grid_index] = rho_response
rho_response = None
# gamma_response = numpy.einsum('dDig,jg,Dg,ij->dg',
# d2mu_dr2[:, :, :, associated_grid_index],
# mu[:, associated_grid_index], drho_dr[:, associated_grid_index], dm0)
# gamma_response += numpy.einsum('dDig,jg,Dg,ij->dg',
# d2mu_dr2[:, :, :, associated_grid_index],
# mu[:, associated_grid_index], drho_dr[:, associated_grid_index], dm0.T)
# gamma_response += numpy.einsum('dig,Djg,Dg,ij->dg',
# dmu_dr[:, :, associated_grid_index],
# dmu_dr[:, :, associated_grid_index], drho_dr[:, associated_grid_index], dm0)
# gamma_response += numpy.einsum('dig,Djg,Dg,ij->dg',
# dmu_dr[:, :, associated_grid_index],
# dmu_dr[:, :, associated_grid_index], drho_dr[:, associated_grid_index], dm0.T)
d2mudr2_dot_drhodr = contract('dDig,Dg->dig',
d2mu_dr2[:, :, :, associated_grid_index], drho_dr[:, associated_grid_index])
gamma_response = contract('dig,ig->dg', d2mudr2_dot_drhodr, dm_dot_mu_and_nu)
d2mudr2_dot_drhodr = None
dm_dot_mu_and_nu = None
dm_dot_dmu_and_dnu = contract('djg,ij->dig',
dmu_dr[:, :, associated_grid_index], dm0 + dm0.T)
dmudr_dot_drhodr = contract('dig,dg->ig',
dmu_dr[:, :, associated_grid_index], drho_dr[:, associated_grid_index])
gamma_response += contract('dig,ig->dg', dm_dot_dmu_and_dnu, dmudr_dot_drhodr)
dmudr_dot_drhodr = None
dm_dot_dmu_and_dnu = None
dgamma_dA_grid_response[i_atom][:, associated_grid_index] = gamma_response
gamma_response = None
else:
assert atom_to_grid_index_map is None
# Here we assume all grids belong to atom i
dm_dot_mu_and_nu = (dm0 + dm0.T) @ mu
drho_dA_grid_response = contract('dig,ig->dg', dmu_dr, dm_dot_mu_and_nu)
d2mudr2_dot_drhodr = contract('dDig,Dg->dig', d2mu_dr2, drho_dr)
dgamma_dA_grid_response = contract('dig,ig->dg', d2mudr2_dot_drhodr, dm_dot_mu_and_nu)
d2mudr2_dot_drhodr = None
dm_dot_mu_and_nu = None
dm_dot_dmu_and_dnu = contract('djg,ij->dig', dmu_dr, dm0 + dm0.T)
dmudr_dot_drhodr = contract('dig,dg->ig', dmu_dr, drho_dr)
dgamma_dA_grid_response += contract('dig,ig->dg', dm_dot_dmu_and_dnu, dmudr_dot_drhodr)
dmudr_dot_drhodr = None
dm_dot_dmu_and_dnu = None
dgamma_dA_grid_response *= 2
return drho_dA_grid_response, dgamma_dA_grid_response
[docs]
def get_d2rhodAdB_d2gammadAdB(mol, grids_coords, dm0):
"""
This function should never be used in practice. It requires crazy amount of memory,
and it's left for debug purpose only. Use the contract function instead.
"""
natm = mol.natm
ngrids = grids_coords.shape[0]
ao = numint.eval_ao(mol, grids_coords, deriv = 3)
rho_drho = numint.eval_rho(mol, ao[:4, :], dm0, xctype = "GGA", hermi = 1, with_lapl = False)
ao = ao.transpose(0,2,1) # order: component, ao, grid
drho = rho_drho[1:4, :]
mu = ao[0, :, :]
dmu_dr = ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(ao)
d3mu_dr3 = get_d3mu_dr3(ao)
aoslices = mol.aoslice_by_atom()
d2rho_dAdB = numpy.zeros([natm, natm, 3, 3, ngrids])
d2gamma_dAdB = numpy.zeros([natm, natm, 3, 3, ngrids])
for i_atom in range(natm):
pi0, pi1 = aoslices[i_atom][2:]
d2rho_dAdB[i_atom, i_atom, :, :, :] += numpy.einsum('dDig,jg,ij->dDg',
d2mu_dr2[:, :, pi0:pi1, :], mu, dm0[pi0:pi1, :])
d2rho_dAdB[i_atom, i_atom, :, :, :] += numpy.einsum('dDig,jg,ij->dDg',
d2mu_dr2[:, :, pi0:pi1, :], mu, dm0[:, pi0:pi1].T)
d2gamma_dAdB[i_atom, i_atom, :, :, :] += numpy.einsum('dDPig,jg,Pg,ij->dDg',
d3mu_dr3[:, :, :, pi0:pi1, :], mu, drho, dm0[pi0:pi1, :])
d2gamma_dAdB[i_atom, i_atom, :, :, :] += numpy.einsum('dDPig,jg,Pg,ij->dDg',
d3mu_dr3[:, :, :, pi0:pi1, :], mu, drho, dm0[:, pi0:pi1].T)
d2gamma_dAdB[i_atom, i_atom, :, :, :] += numpy.einsum('dDig,Pjg,Pg,ij->dDg',
d2mu_dr2[:, :, pi0:pi1, :], dmu_dr, drho, dm0[pi0:pi1, :])
d2gamma_dAdB[i_atom, i_atom, :, :, :] += numpy.einsum('dDig,Pjg,Pg,ij->dDg',
d2mu_dr2[:, :, pi0:pi1, :], dmu_dr, drho, dm0[:, pi0:pi1].T)
for j_atom in range(natm):
pj0, pj1 = aoslices[j_atom][2:]
d2rho_dAdB[i_atom, j_atom, :, :, :] += numpy.einsum('dig,Djg,ij->dDg',
dmu_dr[:, pi0:pi1, :], dmu_dr[:, pj0:pj1, :], dm0[pi0:pi1, pj0:pj1])
d2rho_dAdB[i_atom, j_atom, :, :, :] += numpy.einsum('dig,Djg,ij->dDg',
dmu_dr[:, pi0:pi1, :], dmu_dr[:, pj0:pj1, :], dm0[pj0:pj1, pi0:pi1].T)
d2gamma_dAdB[i_atom, j_atom, :, :, :] += numpy.einsum('dPig,Djg,Pg,ij->dDg',
d2mu_dr2[:, :, pi0:pi1, :], dmu_dr[:, pj0:pj1, :], drho, dm0[pi0:pi1, pj0:pj1])
d2gamma_dAdB[i_atom, j_atom, :, :, :] += numpy.einsum('dPig,Djg,Pg,ij->dDg',
d2mu_dr2[:, :, pi0:pi1, :], dmu_dr[:, pj0:pj1, :], drho, dm0[pj0:pj1, pi0:pi1].T)
d2gamma_dAdB[i_atom, j_atom, :, :, :] += numpy.einsum('dig,DPjg,Pg,ij->dDg',
dmu_dr[:, pi0:pi1, :], d2mu_dr2[:, :, pj0:pj1, :], drho, dm0[pi0:pi1, pj0:pj1])
d2gamma_dAdB[i_atom, j_atom, :, :, :] += numpy.einsum('dig,DPjg,Pg,ij->dDg',
dmu_dr[:, pi0:pi1, :], d2mu_dr2[:, :, pj0:pj1, :], drho, dm0[pj0:pj1, pi0:pi1].T)
d2rho_dAdr = get_d2rho_dAdr_orbital_response(d2mu_dr2, dmu_dr, mu, dm0, aoslices)
d2gamma_dAdB += numpy.einsum('AdPg,BDPg->ABdDg', d2rho_dAdr, d2rho_dAdr)
d2gamma_dAdB *= 2
return d2rho_dAdB, d2gamma_dAdB
[docs]
def contract_d2rhodAdB_d2gammadAdB(d3mu_dr3, d2mu_dr2, dmu_dr, mu, drho_dr, dm0, aoslices, fw_rho, fw_gamma):
assert mu.ndim == 2
nao = mu.shape[0]
ngrids = mu.shape[1]
natm = len(aoslices)
assert d3mu_dr3.shape == (3, 3, 3, nao, ngrids)
assert d2mu_dr2.shape == (3, 3, nao, ngrids)
assert dmu_dr.shape == (3, nao, ngrids)
assert drho_dr.shape == (3, ngrids)
assert dm0.shape == (nao, nao)
drhodr_dot_dmudr = contract('djg,dg->jg', dmu_dr, drho_dr)
d2e_rho_dAdB = numpy.zeros([natm, natm, 3, 3])
d2e_gamma_dAdB = numpy.zeros([natm, natm, 3, 3])
for i_atom in range(natm):
pi0, pi1 = aoslices[i_atom][2:]
nu_dot_dm = dm0[pi0:pi1, :] @ mu
d2rho_dA2 = contract('dDig,ig->dDg', d2mu_dr2[:, :, pi0:pi1, :], nu_dot_dm)
mu_dot_dm = dm0[:, pi0:pi1].T @ mu
d2rho_dA2 += contract('dDig,ig->dDg', d2mu_dr2[:, :, pi0:pi1, :], mu_dot_dm)
d2e_rho_dAdB[i_atom, i_atom, :, :] += contract('dDg,g->dD', d2rho_dA2, fw_rho)
d2rho_dA2 = None
d3mudA2dr_dot_drhodr = contract('dDPig,Pg->dDig', d3mu_dr3[:, :, :, pi0:pi1, :], drho_dr)
d2gamma_dA2 = contract('dDig,ig->dDg', d3mudA2dr_dot_drhodr, nu_dot_dm)
d2gamma_dA2 += contract('dDig,ig->dDg', d3mudA2dr_dot_drhodr, mu_dot_dm)
d3mudA2dr_dot_drhodr = None
nu_dot_dm = None
mu_dot_dm = None
drhodr_dot_dmudr_dot_dm = dm0[pi0:pi1, :] @ drhodr_dot_dmudr
d2gamma_dA2 += contract('dDig,ig->dDg', d2mu_dr2[:, :, pi0:pi1, :], drhodr_dot_dmudr_dot_dm)
drhodr_dot_dmudr_dot_dm = None
drhodr_dot_dnudr_dot_dm = dm0[:, pi0:pi1].T @ drhodr_dot_dmudr
d2gamma_dA2 += contract('dDig,ig->dDg', d2mu_dr2[:, :, pi0:pi1, :], drhodr_dot_dnudr_dot_dm)
drhodr_dot_dnudr_dot_dm = None
d2e_gamma_dAdB[i_atom, i_atom, :, :] += contract('dDg,g->dD', d2gamma_dA2, fw_gamma)
d2gamma_dA2 = None
for j_atom in range(natm):
pj0, pj1 = aoslices[j_atom][2:]
dnudr_dot_dm = contract('djg,ij->dig', dmu_dr[:, pj0:pj1, :], dm0[pi0:pi1, pj0:pj1])
d2rho_dAdB = contract('dig,Dig->dDg', dmu_dr[:, pi0:pi1, :], dnudr_dot_dm)
dmudr_dot_dm = contract('djg,ij->dig', dmu_dr[:, pj0:pj1, :], dm0[pj0:pj1, pi0:pi1].T)
d2rho_dAdB += contract('dig,Dig->dDg', dmu_dr[:, pi0:pi1, :], dmudr_dot_dm)
d2e_rho_dAdB[i_atom, j_atom, :, :] += contract('dDg,g->dD', d2rho_dAdB, fw_rho)
d2rho_dAdB = None
drhodr_dot_d2mudAdr = contract('dDig,Dg->dig', d2mu_dr2[:, :, pi0:pi1, :], drho_dr)
d2gamma_dAdB = contract('dig,Dig->dDg', drhodr_dot_d2mudAdr, dnudr_dot_dm)
dnudr_dot_dm = None
d2gamma_dAdB += contract('dig,Dig->dDg', drhodr_dot_d2mudAdr, dmudr_dot_dm)
dmudr_dot_dm = None
drhodr_dot_d2mudAdr = None
d2gamma_dAdB = contract('dDg,g->dD', d2gamma_dAdB, fw_gamma)
d2e_gamma_dAdB[i_atom, j_atom, :, :] += d2gamma_dAdB
d2e_gamma_dAdB[j_atom, i_atom, :, :] += d2gamma_dAdB.T
d2gamma_dAdB = None
d2rho_dAdr = get_d2rho_dAdr_orbital_response(d2mu_dr2, dmu_dr, mu, dm0, aoslices)
d2e_gamma_dAdB += contract('AdPg,BDPg->ABdD', d2rho_dAdr, d2rho_dAdr * fw_gamma)
return d2e_rho_dAdB + 2 * d2e_gamma_dAdB
def _get_enlc_deriv2(hessobj, mo_coeff, mo_occ, max_memory):
"""
Equation notation follows:
Liang J, Feng X, Liu X, Head-Gordon M. Analytical harmonic vibrational frequencies with
VV10-containing density functionals: Theory, efficient implementation, and
benchmark assessments. J Chem Phys. 2023 May 28;158(20):204109. doi: 10.1063/5.0152838.
"""
mol = hessobj.mol
mf = hessobj.base
mocc = mo_coeff[:,mo_occ>0]
dm0 = 2 * mocc @ mocc.T
grids = mf.nlcgrids
if grids.coords is None:
grids.build()
if numint.libxc.is_nlc(mf.xc):
xc_code = mf.xc
else:
xc_code = mf.nlc
nlc_coefs = mf._numint.nlc_coeff(xc_code)
if len(nlc_coefs) != 1:
raise NotImplementedError('Additive NLC')
nlc_pars, fac = nlc_coefs[0]
kappa_prefactor = nlc_pars[0] * 1.5 * numpy.pi * (9 * numpy.pi)**(-1.0/6.0)
C_in_omega = nlc_pars[1]
beta = 0.03125 * (3.0 / nlc_pars[0]**2)**0.75
# ao = numint.eval_ao(mol, grids.coords, deriv = 3)
# rho_drho = numint.eval_rho(mol, ao, dm0, xctype = "NLC", hermi = 1, with_lapl = False)
ngrids_full = grids.coords.shape[0]
rho_drho = numpy.empty([4, ngrids_full])
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
ao_nbytes_per_grid = ((4*2) * mol.nao + 4) * 8 # factor of 2 from the ao sorting inside numint.eval_ao()
ngrids_per_batch = int(available_cpu_memory / ao_nbytes_per_grid)
if ngrids_per_batch < 16:
raise MemoryError(f"Out of CPU memory for NLC energy second derivative, "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids = {ngrids_full}")
ngrids_per_batch = (ngrids_per_batch + 16 - 1) // 16 * 16
ngrids_per_batch = min(ngrids_per_batch, min_grid_blksize)
for g0 in range(0, ngrids_full, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids_full)
split_grids_coords = grids.coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 1)
split_rho_drho = numint.eval_rho(mol, split_ao, dm0, xctype = "NLC", hermi = 1, with_lapl = False)
rho_drho[:, g0:g1] = split_rho_drho
rho_i = rho_drho[0,:]
rho_nonzero_mask = (rho_i >= NLC_REMOVE_ZERO_RHO_GRID_THRESHOLD)
rho_i = rho_i[rho_nonzero_mask]
nabla_rho_i = rho_drho[1:4, rho_nonzero_mask]
grids_coords = numpy.ascontiguousarray(grids.coords[rho_nonzero_mask, :])
grids_weights = grids.weights[rho_nonzero_mask]
ngrids = grids_coords.shape[0]
gamma_i = nabla_rho_i[0,:]**2 + nabla_rho_i[1,:]**2 + nabla_rho_i[2,:]**2
omega_i = numpy.sqrt(C_in_omega * gamma_i**2 / rho_i**4 + (4.0/3.0*numpy.pi) * rho_i)
kappa_i = kappa_prefactor * rho_i**(1.0/6.0)
U_i = numpy.empty(ngrids)
W_i = numpy.empty(ngrids)
A_i = numpy.empty(ngrids)
B_i = numpy.empty(ngrids)
C_i = numpy.empty(ngrids)
E_i = numpy.empty(ngrids)
libdft.VXC_vv10nlc_hessian_eval_UWABCE(
U_i.ctypes.data_as(ctypes.c_void_p),
W_i.ctypes.data_as(ctypes.c_void_p),
A_i.ctypes.data_as(ctypes.c_void_p),
B_i.ctypes.data_as(ctypes.c_void_p),
C_i.ctypes.data_as(ctypes.c_void_p),
E_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids)
)
domega_drho_i = numpy.empty(ngrids)
domega_dgamma_i = numpy.empty(ngrids)
d2omega_drho2_i = numpy.empty(ngrids)
d2omega_dgamma2_i = numpy.empty(ngrids)
d2omega_drho_dgamma_i = numpy.empty(ngrids)
libdft.VXC_vv10nlc_hessian_eval_omega_derivative(
domega_drho_i.ctypes.data_as(ctypes.c_void_p),
domega_dgamma_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_dgamma2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho_dgamma_i.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
gamma_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_double(C_in_omega),
ctypes.c_int(ngrids)
)
dkappa_drho_i = kappa_prefactor * (1.0/6.0) * rho_i**(-5.0/6.0)
d2kappa_drho2_i = kappa_prefactor * (-5.0/36.0) * rho_i**(-11.0/6.0)
f_rho_i = beta + E_i + rho_i * (dkappa_drho_i * U_i + domega_drho_i * W_i)
f_gamma_i = rho_i * domega_dgamma_i * W_i
f_rho_i = f_rho_i * grids_weights
f_gamma_i = f_gamma_i * grids_weights
aoslices = mol.aoslice_by_atom()
natm = mol.natm
# ao = numint.eval_ao(mol, grids.coords, deriv = 3)
# ao = ao.transpose(0,2,1) # order: component, ao, grid
# ao_nonzero_rho = ao[:, :, rho_nonzero_mask]
# mu = ao_nonzero_rho[0, :, :]
# dmu_dr = ao_nonzero_rho[1:4, :, :]
# d2mu_dr2 = get_d2mu_dr2(ao_nonzero_rho)
# d3mu_dr3 = get_d3mu_dr3(ao_nonzero_rho)
# drho_dA, dgamma_dA = get_drhodA_dgammadA_orbital_response(d2mu_dr2, dmu_dr, mu, nabla_rho_i, dm0, aoslices)
# d2e = contract_d2rhodAdB_d2gammadAdB(d3mu_dr3, d2mu_dr2, dmu_dr, mu, nabla_rho_i, dm0, aoslices,
# f_rho_i, f_gamma_i)
drho_dA = numpy.empty([natm, 3, ngrids], order = "C")
dgamma_dA = numpy.empty([natm, 3, ngrids], order = "C")
d2e = numpy.zeros([natm, natm, 3, 3])
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
ao_nbytes_per_grid = ((20 + 1*2 + 3*2 + 9 + 27) * mol.nao + (3*2 + 9) * mol.natm) * 8
ngrids_per_batch = int(available_cpu_memory / ao_nbytes_per_grid)
if ngrids_per_batch < 16:
raise MemoryError(f"Out of CPU memory for NLC energy second derivative, "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids (nonzero rho) = {ngrids}")
ngrids_per_batch = (ngrids_per_batch + 16 - 1) // 16 * 16
ngrids_per_batch = min(ngrids_per_batch, min_grid_blksize)
for g0 in range(0, ngrids, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids)
split_grids_coords = grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 3)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
d3mu_dr3 = get_d3mu_dr3(split_ao)
split_drho_dr = nabla_rho_i[:, g0:g1]
split_drho_dA, split_dgamma_dA = \
get_drhodA_dgammadA_orbital_response(d2mu_dr2, dmu_dr, mu, split_drho_dr, dm0, aoslices)
drho_dA [:, :, g0:g1] = split_drho_dA
dgamma_dA[:, :, g0:g1] = split_dgamma_dA
split_fw_rho = f_rho_i [g0:g1]
split_fw_gamma = f_gamma_i[g0:g1]
d2e += contract_d2rhodAdB_d2gammadAdB(
d3mu_dr3, d2mu_dr2, dmu_dr, mu, split_drho_dr, dm0, aoslices, split_fw_rho, split_fw_gamma)
split_ao = None
mu = None
dmu_dr = None
d2mu_dr2 = None
d3mu_dr3 = None
split_drho_dA = None
split_dgamma_dA = None
drho_dA = numpy.ascontiguousarray(drho_dA)
dgamma_dA = numpy.ascontiguousarray(dgamma_dA)
f_rho_A_i = numpy.empty([mol.natm, 3, ngrids], order = "C")
f_gamma_A_i = numpy.empty([mol.natm, 3, ngrids], order = "C")
libdft.VXC_vv10nlc_hessian_eval_f_t(
f_rho_A_i.ctypes.data_as(ctypes.c_void_p),
f_gamma_A_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
U_i.ctypes.data_as(ctypes.c_void_p),
W_i.ctypes.data_as(ctypes.c_void_p),
A_i.ctypes.data_as(ctypes.c_void_p),
B_i.ctypes.data_as(ctypes.c_void_p),
C_i.ctypes.data_as(ctypes.c_void_p),
domega_drho_i.ctypes.data_as(ctypes.c_void_p),
domega_dgamma_i.ctypes.data_as(ctypes.c_void_p),
dkappa_drho_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_dgamma2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho_dgamma_i.ctypes.data_as(ctypes.c_void_p),
d2kappa_drho2_i.ctypes.data_as(ctypes.c_void_p),
drho_dA.ctypes.data_as(ctypes.c_void_p),
dgamma_dA.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids),
ctypes.c_int(3 * mol.natm),
)
d2e += contract("Adg,BDg->ABdD", drho_dA, f_rho_A_i * grids_weights)
d2e += contract("Adg,BDg->ABdD", dgamma_dA, f_gamma_A_i * grids_weights)
return d2e
def _get_vxc_deriv1(hessobj, mo_coeff, mo_occ, max_memory):
mol = hessobj.mol
mf = hessobj.base
if hessobj.grids is not None:
grids = hessobj.grids
else:
grids = mf.grids
if grids.coords is None:
grids.build(with_non0tab=True)
nao, nmo = mo_coeff.shape
ni = mf._numint
xctype = ni._xc_type(mf.xc)
aoslices = mol.aoslice_by_atom()
shls_slice = (0, mol.nbas)
ao_loc = mol.ao_loc_nr()
dm0 = mf.make_rdm1(mo_coeff, mo_occ)
v_ip = numpy.zeros((3,nao,nao))
vmat = numpy.zeros((mol.natm,3,nao,nao))
max_memory = max(2000, max_memory-vmat.size*8/1e6)
if xctype == 'LDA':
ao_deriv = 1
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[0], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc_eff(mf.xc, rho, 2, xctype=xctype)[1:3]
wv = weight * vxc[0]
aow = numint._scale_ao(ao[0], wv)
rks_grad._d1_dot_(v_ip, mol, ao[1:4], aow, mask, ao_loc, True)
ao_dm0 = numint._dot_ao_dm(mol, ao[0], dm0, mask, shls_slice, ao_loc)
wf = weight * fxc[0,0]
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
# First order density = rho1 * 2. *2 is not applied because + c.c. in the end
rho1 = numpy.einsum('xpi,pi->xp', ao[1:,:,p0:p1], ao_dm0[:,p0:p1])
wv = wf * rho1
aow = [numint._scale_ao(ao[0], wv[i]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[0], mask, ao_loc, True)
ao_dm0 = aow = None
elif xctype == 'GGA':
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:4], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc_eff(mf.xc, rho, 2, xctype=xctype)[1:3]
wv = weight * vxc
wv[0] *= .5
rks_grad._gga_grad_sum_(v_ip, mol, ao, wv, mask, ao_loc)
ao_dm0 = [numint._dot_ao_dm(mol, ao[i], dm0, mask, shls_slice, ao_loc)
for i in range(4)]
wf = weight * fxc
for ia in range(mol.natm):
dR_rho1 = _make_dR_rho1(ao, ao_dm0, ia, aoslices, xctype)
wv = numpy.einsum('xyg,sxg->syg', wf, dR_rho1)
wv[:,0] *= .5
aow = [numint._scale_ao(ao[:4], wv[i,:4]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[0], mask, ao_loc, True)
ao_dm0 = aow = None
elif xctype == 'MGGA':
_check_mgga_grids(grids)
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:10], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc_eff(mf.xc, rho, 2, xctype=xctype)[1:3]
wv = weight * vxc
wv[0] *= .5
wv[4] *= .5 # for the factor 1/2 in tau
rks_grad._gga_grad_sum_(v_ip, mol, ao, wv, mask, ao_loc)
rks_grad._tau_grad_dot_(v_ip, mol, ao, wv[4], mask, ao_loc, True)
ao_dm0 = [numint._dot_ao_dm(mol, ao[i], dm0, mask, shls_slice, ao_loc) for i in range(4)]
wf = weight * fxc
for ia in range(mol.natm):
dR_rho1 = _make_dR_rho1(ao, ao_dm0, ia, aoslices, xctype)
wv = numpy.einsum('xyg,sxg->syg', wf, dR_rho1)
wv[:,0] *= .5
wv[:,4] *= .25
aow = [numint._scale_ao(ao[:4], wv[i,:4]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[0], mask, ao_loc, True)
for j in range(1, 4):
aow = [numint._scale_ao(ao[j], wv[i,4]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[j], mask, ao_loc, True)
ao_dm0 = aow = None
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmat[ia,:,p0:p1] += v_ip[:,p0:p1]
vmat[ia] = -vmat[ia] - vmat[ia].transpose(0,2,1)
return vmat
def _get_vnlc_deriv1_numerical(hessobj, mo_coeff, mo_occ, max_memory):
"""
Attention: Numerical nlc Fock matrix 1st derivative includes grid response.
"""
mol = hessobj.mol
mf = hessobj.base
mocc = mo_coeff[:,mo_occ>0]
dm0 = numpy.dot(mocc, mocc.T) * 2
nao = mol.nao
vmat = numpy.empty([mol.natm, 3, nao, nao])
def get_nlc_vmat(mol, mf, dm):
ni = mf._numint
if ni.libxc.is_nlc(mf.xc):
xc = mf.xc
else:
assert ni.libxc.is_nlc(mf.nlc)
xc = mf.nlc
mf.nlcgrids.build()
_, _, vnlc = ni.nr_nlc_vxc(mol, mf.nlcgrids, xc, dm)
return vnlc
dx = 1e-3
mol_copy = mol.copy()
mol_copy.verbose = 0
for i_atom in range(mol.natm):
for i_xyz in range(3):
xyz_p = mol.atom_coords()
xyz_p[i_atom, i_xyz] += dx
mol_copy.set_geom_(xyz_p, unit='Bohr')
mol_copy.build()
mf.reset(mol_copy)
vmat_p = get_nlc_vmat(mol_copy, mf, dm0)
xyz_m = mol.atom_coords()
xyz_m[i_atom, i_xyz] -= dx
mol_copy.set_geom_(xyz_m, unit='Bohr')
mol_copy.build()
mf.reset(mol_copy)
vmat_m = get_nlc_vmat(mol_copy, mf, dm0)
vmat[i_atom, i_xyz, :, :] = (vmat_p - vmat_m) / (2 * dx)
mf.reset(mol)
return vmat
[docs]
def get_dweight_dA(mol, grids):
ngrids = grids.coords.shape[0]
assert grids.atm_idx.shape[0] == ngrids
assert grids.quadrature_weights.shape[0] == ngrids
atm_coords = numpy.asarray(mol.atom_coords(), order = "C")
from pyscf.dft.gen_grid import original_becke
assert grids.becke_scheme is original_becke
radii_adjust = grids.radii_adjust
atomic_radii = grids.atomic_radii
if callable(radii_adjust) and atomic_radii is not None:
f_radii_adjust = radii_adjust(mol, atomic_radii)
f_radii_table = numpy.asarray([f_radii_adjust(i, j, 0)
for i in range(mol.natm)
for j in range(mol.natm)])
else:
f_radii_table = numpy.zeros([mol.natm, mol.natm])
dweight_dA = numpy.zeros([mol.natm, 3, ngrids], order = "C")
libdft.VXCbecke_weight_derivative(
dweight_dA.ctypes.data_as(ctypes.c_void_p),
grids.coords.ctypes.data_as(ctypes.c_void_p),
grids.quadrature_weights.ctypes.data_as(ctypes.c_void_p),
atm_coords.ctypes.data_as(ctypes.c_void_p),
f_radii_table.ctypes.data_as(ctypes.c_void_p),
grids.atm_idx.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids),
ctypes.c_int(mol.natm),
)
dweight_dA[grids.atm_idx, 0, numpy.arange(ngrids)] = -numpy.sum(dweight_dA[:, 0, :], axis=0)
dweight_dA[grids.atm_idx, 1, numpy.arange(ngrids)] = -numpy.sum(dweight_dA[:, 1, :], axis=0)
dweight_dA[grids.atm_idx, 2, numpy.arange(ngrids)] = -numpy.sum(dweight_dA[:, 2, :], axis=0)
return dweight_dA
def _get_vnlc_deriv1(hessobj, mo_coeff, mo_occ, max_memory):
"""
Equation notation follows:
Liang J, Feng X, Liu X, Head-Gordon M. Analytical harmonic vibrational frequencies with
VV10-containing density functionals: Theory, efficient implementation, and
benchmark assessments. J Chem Phys. 2023 May 28;158(20):204109. doi: 10.1063/5.0152838.
"""
# Note (Henry Wang 20250428):
# We observed that in several very simple systems, for example H2O2, H2CO, C2H4,
# if we do not include the grid response term, the analytical and numerical Fock matrix
# derivative, although only diff by else than 1e-7 (norm 1), can cause a 1e-3 error in hessian,
# likely because the CPHF converged to a different solution.
grid_response = True
mol = hessobj.mol
mf = hessobj.base
natm = mol.natm
mocc = mo_coeff[:,mo_occ>0]
dm0 = 2 * mocc @ mocc.T
grids = mf.nlcgrids
if grids.coords is None:
grids.build()
if numint.libxc.is_nlc(mf.xc):
xc_code = mf.xc
else:
xc_code = mf.nlc
nlc_coefs = mf._numint.nlc_coeff(xc_code)
if len(nlc_coefs) != 1:
raise NotImplementedError('Additive NLC')
nlc_pars, fac = nlc_coefs[0]
kappa_prefactor = nlc_pars[0] * 1.5 * numpy.pi * (9 * numpy.pi)**(-1.0/6.0)
C_in_omega = nlc_pars[1]
beta = 0.03125 * (3.0 / nlc_pars[0]**2)**0.75
# ao = numint.eval_ao(mol, grids.coords, deriv = 2)
# rho_drho = numint.eval_rho(mol, ao[:4, :], dm0, xctype = "NLC", hermi = 1, with_lapl = False)
ngrids_full = grids.coords.shape[0]
rho_drho = numpy.empty([4, ngrids_full])
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
ao_nbytes_per_grid = ((4*2) * mol.nao + 4) * 8 # factor of 2 from the ao sorting inside numint.eval_ao()
ngrids_per_batch = int(available_cpu_memory / ao_nbytes_per_grid)
if ngrids_per_batch < 16:
raise MemoryError(f"Out of CPU memory for NLC Fock first derivative, "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids = {ngrids_full}")
ngrids_per_batch = (ngrids_per_batch + 16 - 1) // 16 * 16
ngrids_per_batch = min(ngrids_per_batch, min_grid_blksize)
for g0 in range(0, ngrids_full, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids_full)
split_grids_coords = grids.coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 1)
split_rho_drho = numint.eval_rho(mol, split_ao, dm0, xctype = "NLC", hermi = 1, with_lapl = False)
rho_drho[:, g0:g1] = split_rho_drho
rho_i = rho_drho[0,:]
rho_nonzero_mask = (rho_i >= NLC_REMOVE_ZERO_RHO_GRID_THRESHOLD)
rho_i = rho_i[rho_nonzero_mask]
nabla_rho_i = rho_drho[1:4, rho_nonzero_mask]
grids_coords = numpy.ascontiguousarray(grids.coords[rho_nonzero_mask, :])
grids_weights = grids.weights[rho_nonzero_mask]
ngrids = grids_coords.shape[0]
gamma_i = nabla_rho_i[0,:]**2 + nabla_rho_i[1,:]**2 + nabla_rho_i[2,:]**2
omega_i = numpy.sqrt(C_in_omega * gamma_i**2 / rho_i**4 + (4.0/3.0*numpy.pi) * rho_i)
kappa_i = kappa_prefactor * rho_i**(1.0/6.0)
U_i = numpy.empty(ngrids)
W_i = numpy.empty(ngrids)
A_i = numpy.empty(ngrids)
B_i = numpy.empty(ngrids)
C_i = numpy.empty(ngrids)
E_i = numpy.empty(ngrids)
libdft.VXC_vv10nlc_hessian_eval_UWABCE(
U_i.ctypes.data_as(ctypes.c_void_p),
W_i.ctypes.data_as(ctypes.c_void_p),
A_i.ctypes.data_as(ctypes.c_void_p),
B_i.ctypes.data_as(ctypes.c_void_p),
C_i.ctypes.data_as(ctypes.c_void_p),
E_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids)
)
domega_drho_i = numpy.empty(ngrids)
domega_dgamma_i = numpy.empty(ngrids)
d2omega_drho2_i = numpy.empty(ngrids)
d2omega_dgamma2_i = numpy.empty(ngrids)
d2omega_drho_dgamma_i = numpy.empty(ngrids)
libdft.VXC_vv10nlc_hessian_eval_omega_derivative(
domega_drho_i.ctypes.data_as(ctypes.c_void_p),
domega_dgamma_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_dgamma2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho_dgamma_i.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
gamma_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_double(C_in_omega),
ctypes.c_int(ngrids)
)
dkappa_drho_i = kappa_prefactor * (1.0/6.0) * rho_i**(-5.0/6.0)
d2kappa_drho2_i = kappa_prefactor * (-5.0/36.0) * rho_i**(-11.0/6.0)
f_rho_i = beta + E_i + rho_i * (dkappa_drho_i * U_i + domega_drho_i * W_i)
f_gamma_i = rho_i * domega_dgamma_i * W_i
aoslices = mol.aoslice_by_atom()
if grid_response:
assert grids.atm_idx.shape[0] == grids.coords.shape[0]
grid_to_atom_index_map = grids.atm_idx[rho_nonzero_mask]
atom_to_grid_index_map = [numpy.where(grid_to_atom_index_map == i_atom)[0] for i_atom in range(natm)]
# ao = numint.eval_ao(mol, grids.coords, deriv = 2)
# ao = ao.transpose(0,2,1) # order: component, ao, grid
# ao_nonzero_rho = ao[:,:,rho_nonzero_mask]
# mu = ao_nonzero_rho[0, :, :]
# dmu_dr = ao_nonzero_rho[1:4, :, :]
# d2mu_dr2 = get_d2mu_dr2(ao_nonzero_rho)
# drho_dA, dgamma_dA = get_drhodA_dgammadA_orbital_response(d2mu_dr2, dmu_dr, mu, nabla_rho_i, dm0, aoslices)
# if grid_response:
# drho_dA_grid_response, dgamma_dA_grid_response = \
# get_drhodA_dgammadA_grid_response(d2mu_dr2, dmu_dr, mu, nabla_rho_i, dm0,
# atom_to_grid_index_map = atom_to_grid_index_map)
# drho_dA += drho_dA_grid_response
# dgamma_dA += dgamma_dA_grid_response
# drho_dA_grid_response = None
# dgamma_dA_grid_response = None
drho_dA = numpy.empty([natm, 3, ngrids], order = "C")
dgamma_dA = numpy.empty([natm, 3, ngrids], order = "C")
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
ao_nbytes_per_grid = ((10 + 1*2 + 3*2 + 9) * mol.nao + (3*2) * mol.natm) * 8
ngrids_per_batch = int(available_cpu_memory / ao_nbytes_per_grid)
if ngrids_per_batch < 16:
raise MemoryError(f"Out of CPU memory for NLC Fock first derivative, "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids (nonzero rho) = {ngrids}")
ngrids_per_batch = (ngrids_per_batch + 16 - 1) // 16 * 16
ngrids_per_batch = min(ngrids_per_batch, min_grid_blksize)
for g0 in range(0, ngrids, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids)
split_grids_coords = grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_drho_dr = nabla_rho_i[:, g0:g1]
split_drho_dA, split_dgamma_dA = \
get_drhodA_dgammadA_orbital_response(d2mu_dr2, dmu_dr, mu, split_drho_dr, dm0, aoslices)
drho_dA [:, :, g0:g1] = split_drho_dA
dgamma_dA[:, :, g0:g1] = split_dgamma_dA
split_drho_dA = None
split_dgamma_dA = None
if grid_response:
for i_atom in range(natm):
associated_grid_index = atom_to_grid_index_map[i_atom]
associated_grids_coords = grids_coords[associated_grid_index, :]
ngrids_per_atom = associated_grids_coords.shape[0]
associated_drho_dr = nabla_rho_i[:, associated_grid_index]
drho_dA_grid_response = numpy.empty([3, ngrids_per_atom])
dgamma_dA_grid_response = numpy.empty([3, ngrids_per_atom])
for g0 in range(0, ngrids_per_atom, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids_per_atom)
split_grids_coords = associated_grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_drho_dr = associated_drho_dr[:, g0:g1]
split_drho_dA_grid_response, split_dgamma_dA_grid_response = \
get_drhodA_dgammadA_grid_response(d2mu_dr2, dmu_dr, mu, split_drho_dr, dm0, i_atom = i_atom)
drho_dA_grid_response [:, g0:g1] = split_drho_dA_grid_response
dgamma_dA_grid_response[:, g0:g1] = split_dgamma_dA_grid_response
drho_dA [i_atom][:, associated_grid_index] += drho_dA_grid_response
dgamma_dA[i_atom][:, associated_grid_index] += dgamma_dA_grid_response
drho_dA_grid_response = None
dgamma_dA_grid_response = None
drho_dA = numpy.ascontiguousarray(drho_dA)
dgamma_dA = numpy.ascontiguousarray(dgamma_dA)
f_rho_A_i = numpy.empty([natm, 3, ngrids], order = "C")
f_gamma_A_i = numpy.empty([natm, 3, ngrids], order = "C")
libdft.VXC_vv10nlc_hessian_eval_f_t(
f_rho_A_i.ctypes.data_as(ctypes.c_void_p),
f_gamma_A_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
U_i.ctypes.data_as(ctypes.c_void_p),
W_i.ctypes.data_as(ctypes.c_void_p),
A_i.ctypes.data_as(ctypes.c_void_p),
B_i.ctypes.data_as(ctypes.c_void_p),
C_i.ctypes.data_as(ctypes.c_void_p),
domega_drho_i.ctypes.data_as(ctypes.c_void_p),
domega_dgamma_i.ctypes.data_as(ctypes.c_void_p),
dkappa_drho_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_dgamma2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho_dgamma_i.ctypes.data_as(ctypes.c_void_p),
d2kappa_drho2_i.ctypes.data_as(ctypes.c_void_p),
drho_dA.ctypes.data_as(ctypes.c_void_p),
dgamma_dA.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids),
ctypes.c_int(3 * natm),
)
drho_dA = None
dgamma_dA = None
vmat = numpy.zeros([natm, 3, mol.nao, mol.nao])
# ao = numint.eval_ao(mol, grids.coords, deriv = 2)
# ao = ao.transpose(0,2,1) # order: component, ao, grid
# ao_nonzero_rho = ao[:,:,rho_nonzero_mask]
# mu = ao_nonzero_rho[0, :, :]
# dmu_dr = ao_nonzero_rho[1:4, :, :]
# d2mu_dr2 = get_d2mu_dr2(ao_nonzero_rho)
# d2rho_dAdr = get_d2rho_dAdr_orbital_response(d2mu_dr2, dmu_dr, mu, dm0, aoslices)
# if grid_response:
# d2rho_dAdr_grid_response = get_d2rho_dAdr_grid_response(d2mu_dr2, dmu_dr, mu, dm0,
# atom_to_grid_index_map = atom_to_grid_index_map)
# d2rho_dAdr += d2rho_dAdr_grid_response
# d2rho_dAdr_grid_response = None
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
ao_nbytes_per_grid = ((10 + 1*2 + 3*2 + 9) * mol.nao + (9*2)) * 8
ngrids_per_batch = int(available_cpu_memory / ao_nbytes_per_grid)
if ngrids_per_batch < 16:
raise MemoryError(f"Out of CPU memory for NLC Fock first derivative, "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids (nonzero rho) = {ngrids}")
ngrids_per_batch = (ngrids_per_batch + 16 - 1) // 16 * 16
ngrids_per_batch = min(ngrids_per_batch, min_grid_blksize)
for i_atom in range(natm):
aoslice_one_atom = [aoslices[i_atom]]
d2rho_dAdr = numpy.empty([3, 3, ngrids])
for g0 in range(0, ngrids, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids)
split_grids_coords = grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_drho_dr = nabla_rho_i[:, g0:g1]
split_d2rho_dAdr = get_d2rho_dAdr_orbital_response(d2mu_dr2, dmu_dr, mu, dm0, aoslice_one_atom)
d2rho_dAdr[:, :, g0:g1] = split_d2rho_dAdr
split_d2rho_dAdr = None
if grid_response:
associated_grid_index = atom_to_grid_index_map[i_atom]
associated_grids_coords = grids_coords[associated_grid_index, :]
ngrids_per_atom = associated_grids_coords.shape[0]
d2rho_dAdr_grid_response = numpy.empty([3, 3, ngrids_per_atom])
for g0 in range(0, ngrids_per_atom, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids_per_atom)
split_grids_coords = associated_grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_d2rho_dAdr_grid_response = \
get_d2rho_dAdr_grid_response(d2mu_dr2, dmu_dr, mu, dm0, i_atom = i_atom)
d2rho_dAdr_grid_response[:, :, g0:g1] = split_d2rho_dAdr_grid_response
d2rho_dAdr[:, :, associated_grid_index] += d2rho_dAdr_grid_response
split_d2rho_dAdr_grid_response = None
for g0 in range(0, ngrids, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids)
split_grids_coords = grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_drho_dr = nabla_rho_i[:, g0:g1]
# # w_i 2 f_i^\gamma \nabla_A \nabla\rho \cdot \nabla(\phi_\mu \phi_nu)_i
# vmat[i_atom, :, :, :] += 2 * numpy.einsum(
# 'dDg,Dig,jg,g->dij', d2rho_dAdr[i_atom, :, :, :], dmu_dr, mu, f_gamma_i * grids_weights)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum(
# 'dDg,Dig,jg,g->dji', d2rho_dAdr[i_atom, :, :, :], dmu_dr, mu, f_gamma_i * grids_weights)
d2rhodAdr_dot_dmudr = contract('dDg,Dig->dig', d2rho_dAdr[:, :, g0:g1], dmu_dr)
dF = contract('dig,jg->dij', d2rhodAdr_dot_dmudr, mu * f_gamma_i[g0:g1] * grids_weights[g0:g1])
d2rhodAdr_dot_dmudr = None
# # w_i 2 (\nabla\rho)_i \cdot (\nabla(\phi_\mu \phi_nu))_i f_i^{\gamma, A}
# vmat[i_atom, :, :, :] += 2 * numpy.einsum(
# 'dg,Dig,jg,Dg->dij', f_gamma_A_i[i_atom, :, :], dmu_dr, mu, nabla_rho_i * grids_weights)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum(
# 'dg,Dig,jg,Dg->dji', f_gamma_A_i[i_atom, :, :], dmu_dr, mu, nabla_rho_i * grids_weights)
f_gamma_A_i_mu = contract('dg,ig->dig', f_gamma_A_i[i_atom, :, g0:g1], mu)
drhodr_dot_dmudr = contract('dig,dg->ig', dmu_dr, split_drho_dr * grids_weights[g0:g1])
dF += contract('dig,jg->dij', f_gamma_A_i_mu, drhodr_dot_dmudr)
drhodr_dot_dmudr = None
f_gamma_A_i_mu = None
dF += dF.transpose(0,2,1)
dF *= 2
# # w_i \phi_{\mu i} \phi_{\nu i} f_i^{\rho, A}
# vmat[i_atom, :, :, :] += numpy.einsum('dg,ig,jg,g->dij', f_rho_A_i[i_atom, :, :], mu, mu, grids_weights)
f_rho_A_i_mu = contract('dg,ig->dig', f_rho_A_i[i_atom, :, g0:g1], mu)
dF += contract('dig,jg->dij', f_rho_A_i_mu, mu * grids_weights[g0:g1])
f_rho_A_i_mu = None
vmat[i_atom, :, :, :] += dF
dF = None
p0, p1 = aoslices[i_atom][2:]
# # w_i f_i^\rho \nabla_A (\phi_\mu \phi_nu)_i
# vmat[i_atom, :, p0:p1, :] += numpy.einsum(
# 'dig,jg->dij', -dmu_dr[:, p0:p1, :], mu * f_rho_i * grids_weights)
# vmat[i_atom, :, :, p0:p1] += numpy.einsum(
# 'dig,jg->dji', -dmu_dr[:, p0:p1, :], mu * f_rho_i * grids_weights)
f_rho_dmudA_nu = contract('dig,jg->dij', -dmu_dr[:, p0:p1, :], mu * f_rho_i[g0:g1] * grids_weights[g0:g1])
# # w_i 2 f_i^\gamma \nabla\rho \cdot \nabla_A \nabla(\phi_\mu \phi_nu)_i
# vmat[i_atom, :, p0:p1, :] += 2 * numpy.einsum(
# 'dDig,jg,Dg->dij', -d2mu_dr2[:, :, p0:p1, :], mu, nabla_rho_i * f_gamma_i * grids_weights)
# vmat[i_atom, :, :, p0:p1] += 2 * numpy.einsum(
# 'dDig,jg,Dg->dji', -d2mu_dr2[:, :, p0:p1, :], mu, nabla_rho_i * f_gamma_i * grids_weights)
# vmat[i_atom, :, p0:p1, :] += 2 * numpy.einsum(
# 'dig,Djg,Dg->dij', -dmu_dr[:, p0:p1, :], dmu_dr, nabla_rho_i * f_gamma_i * grids_weights)
# vmat[i_atom, :, :, p0:p1] += 2 * numpy.einsum(
# 'dig,Djg,Dg->dji', -dmu_dr[:, p0:p1, :], dmu_dr, nabla_rho_i * f_gamma_i * grids_weights)
mu_dot_drhodr = contract('ig,dg->dig', mu, split_drho_dr * f_gamma_i[g0:g1] * grids_weights[g0:g1])
f_gamma_d2mudr2_nu = contract('dDig,Djg->dij', -d2mu_dr2[:, :, p0:p1, :], mu_dot_drhodr)
mu_dot_drhodr = None
dmudr_dot_drhodr = contract('dig,dg->ig', dmu_dr, split_drho_dr * f_gamma_i[g0:g1] * grids_weights[g0:g1])
f_gamma_dmudr_dnudr = contract('dig,jg->dij', -dmu_dr[:, p0:p1, :], dmudr_dot_drhodr)
dmudr_dot_drhodr = None
dF_ao = f_rho_dmudA_nu + 2 * (f_gamma_d2mudr2_nu + f_gamma_dmudr_dnudr)
f_rho_dmudA_nu = None
f_gamma_d2mudr2_nu = None
f_gamma_dmudr_dnudr = None
vmat[i_atom, :, p0:p1, :] += dF_ao
vmat[i_atom, :, :, p0:p1] += dF_ao.transpose(0,2,1)
dF_ao = None
d2rho_dAdr = None
if grid_response:
associated_grid_index = atom_to_grid_index_map[i_atom]
associated_grids_coords = grids_coords[associated_grid_index, :]
ngrids_per_atom = associated_grids_coords.shape[0]
associated_drho_dr = nabla_rho_i[:, associated_grid_index]
fw_rho_associated_grids = f_rho_i[associated_grid_index] * grids_weights[associated_grid_index]
fw_gamma_associated_grids = f_gamma_i[associated_grid_index] * grids_weights[associated_grid_index]
for g0 in range(0, ngrids_per_atom, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids_per_atom)
split_grids_coords = associated_grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_drho_dr = associated_drho_dr[:, g0:g1]
# # w_i f_i^\rho \nabla_A (\phi_\mu \phi_nu)_i
# vmat[i_atom, :, :, :] += numpy.einsum('dig,jg->dij',
# dmu_dr[:, :, associated_grid_index],
# mu[:, associated_grid_index] * fw_rho_associated_grids)
# vmat[i_atom, :, :, :] += numpy.einsum('dig,jg->dji',
# dmu_dr[:, :, associated_grid_index],
# mu[:, associated_grid_index] * fw_rho_associated_grids)
f_rho_dmudA_nu = contract('dig,jg->dij', dmu_dr, mu * fw_rho_associated_grids[g0:g1])
# # w_i 2 f_i^\gamma \nabla\rho \cdot \nabla_A \nabla(\phi_\mu \phi_nu)_i
# vmat[i_atom, :, :, :] += 2 * numpy.einsum('dDig,jg,Dg->dij',
# d2mu_dr2[:, :, :, associated_grid_index], mu[:, associated_grid_index],
# nabla_rho_i[:, associated_grid_index] * fw_gamma_associated_grids)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum('dDig,jg,Dg->dji',
# d2mu_dr2[:, :, :, associated_grid_index], mu[:, associated_grid_index],
# nabla_rho_i[:, associated_grid_index] * fw_gamma_associated_grids)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum('dig,Djg,Dg->dij',
# dmu_dr[:, :, associated_grid_index], dmu_dr[:, :, associated_grid_index],
# nabla_rho_i[:, associated_grid_index] * fw_gamma_associated_grids)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum('dig,Djg,Dg->dji',
# dmu_dr[:, :, associated_grid_index], dmu_dr[:, :, associated_grid_index],
# nabla_rho_i[:, associated_grid_index] * fw_gamma_associated_grids)
d2mudr2_dot_drhodr = contract('dDig,Dg->dig',
d2mu_dr2, split_drho_dr * fw_gamma_associated_grids[g0:g1])
f_gamma_d2mudr2_nu = contract('dig,jg->dij', d2mudr2_dot_drhodr, mu)
d2mudr2_dot_drhodr = None
dmudr_dot_drhodr = contract('dig,dg->ig', dmu_dr, split_drho_dr * fw_gamma_associated_grids[g0:g1])
f_gamma_dmudr_dnudr = contract('dig,jg->dij', dmu_dr, dmudr_dot_drhodr)
dmudr_dot_drhodr = None
dF_ao = f_rho_dmudA_nu + 2 * (f_gamma_d2mudr2_nu + f_gamma_dmudr_dnudr)
f_rho_dmudA_nu = None
f_gamma_d2mudr2_nu = None
f_gamma_dmudr_dnudr = None
dF_ao += dF_ao.transpose(0,2,1)
vmat[i_atom, :, :, :] += dF_ao
dF_ao = None
if grid_response:
E_Bgr_i = numpy.empty([natm, 3, ngrids], order = "C")
U_Bgr_i = numpy.empty([natm, 3, ngrids], order = "C")
W_Bgr_i = numpy.empty([natm, 3, ngrids], order = "C")
libdft.VXC_vv10nlc_hessian_eval_EUW_grid_response(
E_Bgr_i.ctypes.data_as(ctypes.c_void_p),
U_Bgr_i.ctypes.data_as(ctypes.c_void_p),
W_Bgr_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
grid_to_atom_index_map.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids),
ctypes.c_int(natm),
)
grids_weights_1 = get_dweight_dA(mol, grids)
grids_weights_1 = grids_weights_1[:, :, rho_nonzero_mask]
grids_weights_1 = numpy.ascontiguousarray(grids_weights_1)
E_Bw_i = numpy.empty([natm, 3, ngrids], order = "C")
U_Bw_i = numpy.empty([natm, 3, ngrids], order = "C")
W_Bw_i = numpy.empty([natm, 3, ngrids], order = "C")
libdft.VXC_vv10nlc_hessian_eval_EUW_with_weight1(
E_Bw_i.ctypes.data_as(ctypes.c_void_p),
U_Bw_i.ctypes.data_as(ctypes.c_void_p),
W_Bw_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights_1.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids),
ctypes.c_int(natm * 3),
)
f_rho_grid_response_i = (E_Bw_i + E_Bgr_i) \
+ ((U_Bw_i + U_Bgr_i) * dkappa_drho_i + (W_Bw_i + W_Bgr_i) * domega_drho_i) * rho_i
f_gamma_grid_response_i = (W_Bw_i + W_Bgr_i) * domega_dgamma_i * rho_i
E_Bw_i = None
U_Bw_i = None
W_Bw_i = None
E_Bgr_i = None
U_Bgr_i = None
W_Bgr_i = None
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
ao_nbytes_per_grid = ((4 + 1*2 + 3*2) * mol.nao) * 8
ngrids_per_batch = int(available_cpu_memory / ao_nbytes_per_grid)
if ngrids_per_batch < 16:
raise MemoryError(f"Out of CPU memory for NLC Fock first derivative, "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids (nonzero rho) = {ngrids}")
ngrids_per_batch = (ngrids_per_batch + 16 - 1) // 16 * 16
ngrids_per_batch = min(ngrids_per_batch, min_grid_blksize)
for g0 in range(0, ngrids, ngrids_per_batch):
g1 = min(g0 + ngrids_per_batch, ngrids)
split_grids_coords = grids_coords[g0:g1, :]
split_ao = numint.eval_ao(mol, split_grids_coords, deriv = 2)
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
mu = split_ao[0, :, :]
dmu_dr = split_ao[1:4, :, :]
d2mu_dr2 = get_d2mu_dr2(split_ao)
split_drho_dr = nabla_rho_i[:, g0:g1]
for i_atom in range(natm):
# # \nabla_A w_i term
# vmat[i_atom, :, :, :] += numpy.einsum(
# 'dg,ig,jg->dij', grids_weights_1[i_atom, :, :], mu, mu * f_rho_i)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum(
# 'dg,Dig,jg,Dg->dij', grids_weights_1[i_atom, :, :], dmu_dr, mu, nabla_rho_i * f_gamma_i)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum(
# 'dg,Dig,jg,Dg->dji', grids_weights_1[i_atom, :, :], dmu_dr, mu, nabla_rho_i * f_gamma_i)
dwdr_dot_mu = contract('dg,ig->dig', grids_weights_1[i_atom, :, g0:g1], mu)
f_rho_dwdr = contract('dig,jg->dij', dwdr_dot_mu, mu * f_rho_i[g0:g1])
dmudr_dot_drhodr = contract('dig,dg->ig', dmu_dr, split_drho_dr * f_gamma_i[g0:g1])
f_gamma_dwdr = contract('dig,jg->dij', dwdr_dot_mu, dmudr_dot_drhodr)
dmudr_dot_drhodr = None
dwdr_dot_mu = None
# # E_i^{Aw} and E_i^{Agr} terms combined
# vmat[i_atom, :, :, :] += numpy.einsum(
# 'dg,ig,jg->dij', f_rho_grid_response_i[i_atom, :, :], mu, mu * grids_weights)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum('dg,Dig,jg,Dg->dij',
# f_gamma_grid_response_i[i_atom, :, :], dmu_dr, mu, nabla_rho_i * grids_weights)
# vmat[i_atom, :, :, :] += 2 * numpy.einsum('dg,Dig,jg,Dg->dji',
# f_gamma_grid_response_i[i_atom, :, :], dmu_dr, mu, nabla_rho_i * grids_weights)
dfrhodr_dot_mu = contract('dg,ig->dig', f_rho_grid_response_i[i_atom, :, g0:g1], mu)
f_rho_dwdr += contract('dig,jg->dij', dfrhodr_dot_mu, mu * grids_weights[g0:g1])
dfrhodr_dot_mu = None
dfgammadr_dot_mu = contract('dg,ig->dig', f_gamma_grid_response_i[i_atom, :, g0:g1], mu)
dmudr_dot_drhodr = contract('dig,dg->ig', dmu_dr, split_drho_dr * grids_weights[g0:g1])
f_gamma_dwdr += contract('dig,jg->dij', dfgammadr_dot_mu, dmudr_dot_drhodr)
dmudr_dot_drhodr = None
dfgammadr_dot_mu = None
f_gamma_dwdr += f_gamma_dwdr.transpose(0,2,1)
dF_ao = f_rho_dwdr + 2 * f_gamma_dwdr
f_rho_dwdr = None
f_gamma_dwdr = None
vmat[i_atom, :, :, :] += dF_ao
dF_ao = None
return vmat
[docs]
def get_vnlc_resp(mf, mol, mo_coeff, mo_occ, dm1s, max_memory):
"""
Equation notation follows:
Liang J, Feng X, Liu X, Head-Gordon M. Analytical harmonic vibrational frequencies with
VV10-containing density functionals: Theory, efficient implementation, and
benchmark assessments. J Chem Phys. 2023 May 28;158(20):204109. doi: 10.1063/5.0152838.
mo_coeff, mo_occ are 0-th order
dm1s is first order
TODO: check the effect of different grid, using mf.nlcgrids right now
"""
if isinstance(mo_coeff, numpy.ndarray) and mo_coeff.ndim == 2:
mocc = mo_coeff[:,mo_occ>0]
mo_occ = mo_occ[mo_occ > 0]
dm0 = (mocc * mo_occ) @ mocc.T
else:
assert mo_coeff[0].ndim == 2 # unrestricted case
assert len(mo_coeff) == 2
assert len(mo_occ) == 2
mocc_a = mo_coeff[0][:, mo_occ[0] > 0]
mocc_b = mo_coeff[1][:, mo_occ[1] > 0]
mo_occ_a = mo_occ[0][mo_occ[0] > 0]
mo_occ_b = mo_occ[1][mo_occ[1] > 0]
dm0 = (mocc_a * mo_occ_a) @ mocc_a.T + (mocc_b * mo_occ_b) @ mocc_b.T
dm0 = numpy.asarray(dm0.real, order='C')
output_in_2d = False
if dm1s.ndim == 2:
assert dm1s.shape == (mol.nao, mol.nao)
dm1s = dm1s.reshape((1, mol.nao, mol.nao))
output_in_2d = True
assert dm1s.ndim == 3
grids = mf.nlcgrids
if grids.coords is None:
grids.build()
n_dm1 = dm1s.shape[0]
ni = mf._numint
if numint.libxc.is_nlc(mf.xc):
xc_code = mf.xc
else:
xc_code = mf.nlc
nlc_coefs = ni.nlc_coeff(xc_code)
if len(nlc_coefs) != 1:
raise NotImplementedError('Additive NLC')
nlc_pars, fac = nlc_coefs[0]
kappa_prefactor = nlc_pars[0] * 1.5 * numpy.pi * (9 * numpy.pi)**(-1.0/6.0)
C_in_omega = nlc_pars[1]
# ao = numint.eval_ao(mol, grids.coords, deriv = 1)
# rho_drho = numint.eval_rho(mol, ao, dm0, xctype = "NLC", hermi = 1, with_lapl = False)
ngrids_full = grids.coords.shape[0]
rho_drho = numpy.empty([4, ngrids_full])
g1 = 0
for split_ao, ao_mask_index, split_weights, split_coords in ni.block_loop(mol, grids, mol.nao, 1, max_memory):
g0, g1 = g1, g1 + split_weights.size
rho_drho[:, g0:g1] = numint.eval_rho(mol, split_ao, dm0, xctype = "NLC", hermi = 1)
dm0 = None
rho_i = rho_drho[0,:]
rho_nonzero_mask = (rho_i >= NLC_REMOVE_ZERO_RHO_GRID_THRESHOLD)
rho_i = rho_i[rho_nonzero_mask]
nabla_rho_i = rho_drho[1:4, rho_nonzero_mask]
grids_coords = numpy.ascontiguousarray(grids.coords[rho_nonzero_mask, :])
grids_weights = grids.weights[rho_nonzero_mask]
ngrids = grids_coords.shape[0]
gamma_i = nabla_rho_i[0,:]**2 + nabla_rho_i[1,:]**2 + nabla_rho_i[2,:]**2
omega_i = numpy.sqrt(C_in_omega * gamma_i**2 / rho_i**4 + (4.0/3.0*numpy.pi) * rho_i)
kappa_i = kappa_prefactor * rho_i**(1.0/6.0)
U_i = numpy.empty(ngrids)
W_i = numpy.empty(ngrids)
A_i = numpy.empty(ngrids)
B_i = numpy.empty(ngrids)
C_i = numpy.empty(ngrids)
E_i = numpy.empty(ngrids) # Not used
libdft.VXC_vv10nlc_hessian_eval_UWABCE(
U_i.ctypes.data_as(ctypes.c_void_p),
W_i.ctypes.data_as(ctypes.c_void_p),
A_i.ctypes.data_as(ctypes.c_void_p),
B_i.ctypes.data_as(ctypes.c_void_p),
C_i.ctypes.data_as(ctypes.c_void_p),
E_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids)
)
E_i = None
domega_drho_i = numpy.empty(ngrids)
domega_dgamma_i = numpy.empty(ngrids)
d2omega_drho2_i = numpy.empty(ngrids)
d2omega_dgamma2_i = numpy.empty(ngrids)
d2omega_drho_dgamma_i = numpy.empty(ngrids)
libdft.VXC_vv10nlc_hessian_eval_omega_derivative(
domega_drho_i.ctypes.data_as(ctypes.c_void_p),
domega_dgamma_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_dgamma2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho_dgamma_i.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
gamma_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_double(C_in_omega),
ctypes.c_int(ngrids)
)
dkappa_drho_i = kappa_prefactor * (1.0/6.0) * rho_i**(-5.0/6.0)
d2kappa_drho2_i = kappa_prefactor * (-5.0/36.0) * rho_i**(-11.0/6.0)
f_gamma_i = rho_i * domega_dgamma_i * W_i
# ao = numint.eval_ao(mol, grids.coords, deriv = 1)
# rho_drho_t = numpy.empty([n_dm1, 4, ngrids])
# for i_dm in range(n_dm1):
# dm1 = dm1s[i_dm, :, :]
# rho_drho_1 = numint.eval_rho(mol, ao, dm1, xctype = "NLC", hermi = 0, with_lapl = False)
# rho_drho_t[i_dm, :, :] = rho_drho_1[:, rho_nonzero_mask]
vmat = numpy.zeros([n_dm1, mol.nao, mol.nao])
mem_now = lib.current_memory()[0]
available_cpu_memory = max(16e3, max_memory * 0.5 - mem_now) * 1e6
fxc_nbytes_per_dm1 = ((1*6 + 3*2) * ngrids + (1*2 + 3*2) * ngrids_full) * 8
ndm1_per_batch = int(available_cpu_memory / fxc_nbytes_per_dm1)
if ndm1_per_batch < 6:
raise MemoryError(f"Out of CPU memory for NLC response (orbital hessian), "
f"available cpu memory = {available_cpu_memory}"
f" bytes, nao = {mol.nao}, natm = {mol.natm}, ngrids (nonzero rho) = {ngrids}")
ndm1_per_batch = (ndm1_per_batch + 6 - 1) // 6 * 6
for i_dm1_batch in range(0, n_dm1, ndm1_per_batch):
n_dm1_batch = min(ndm1_per_batch, n_dm1 - i_dm1_batch)
rho_drho_t = numpy.empty([n_dm1_batch, 4, ngrids_full])
g1 = 0
for split_ao, ao_mask_index, split_weights, split_coords in ni.block_loop(mol, grids, mol.nao, 1, max_memory):
g0, g1 = g1, g1 + split_weights.size
for i_dm in range(n_dm1_batch):
dm1_subset = dm1s[i_dm + i_dm1_batch, :, :]
rho_drho_t[i_dm, :, g0:g1] = numint.eval_rho(mol, split_ao, dm1_subset, xctype = "NLC", hermi = 0)
dm1_subset = None
rho_drho_t = rho_drho_t[:, :, rho_nonzero_mask]
rho_t_i = rho_drho_t[:, 0, :]
nabla_rho_t_i = rho_drho_t[:, 1:4, :]
gamma_t_i = nabla_rho_i[0, :] * nabla_rho_t_i[:, 0, :] \
+ nabla_rho_i[1, :] * nabla_rho_t_i[:, 1, :] \
+ nabla_rho_i[2, :] * nabla_rho_t_i[:, 2, :]
gamma_t_i *= 2 # Account for the factor of 2 before gamma_j^t term in equation (22)
rho_drho_t = None
rho_t_i = numpy.ascontiguousarray(rho_t_i)
gamma_t_i = numpy.ascontiguousarray(gamma_t_i)
f_rho_t_i = numpy.empty([n_dm1_batch, ngrids], order = "C")
f_gamma_t_i = numpy.empty([n_dm1_batch, ngrids], order = "C")
libdft.VXC_vv10nlc_hessian_eval_f_t(
f_rho_t_i.ctypes.data_as(ctypes.c_void_p),
f_gamma_t_i.ctypes.data_as(ctypes.c_void_p),
grids_coords.ctypes.data_as(ctypes.c_void_p),
grids_weights.ctypes.data_as(ctypes.c_void_p),
rho_i.ctypes.data_as(ctypes.c_void_p),
omega_i.ctypes.data_as(ctypes.c_void_p),
kappa_i.ctypes.data_as(ctypes.c_void_p),
U_i.ctypes.data_as(ctypes.c_void_p),
W_i.ctypes.data_as(ctypes.c_void_p),
A_i.ctypes.data_as(ctypes.c_void_p),
B_i.ctypes.data_as(ctypes.c_void_p),
C_i.ctypes.data_as(ctypes.c_void_p),
domega_drho_i.ctypes.data_as(ctypes.c_void_p),
domega_dgamma_i.ctypes.data_as(ctypes.c_void_p),
dkappa_drho_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_dgamma2_i.ctypes.data_as(ctypes.c_void_p),
d2omega_drho_dgamma_i.ctypes.data_as(ctypes.c_void_p),
d2kappa_drho2_i.ctypes.data_as(ctypes.c_void_p),
rho_t_i.ctypes.data_as(ctypes.c_void_p),
gamma_t_i.ctypes.data_as(ctypes.c_void_p),
ctypes.c_int(ngrids),
ctypes.c_int(n_dm1_batch),
)
rho_t_i = None
gamma_t_i = None
fxc_rho = f_rho_t_i * grids_weights
f_rho_t_i = None
fxc_gamma = contract("dg,tg->tdg", nabla_rho_i, f_gamma_t_i)
f_gamma_t_i = None
fxc_gamma += nabla_rho_t_i * f_gamma_i
nabla_rho_t_i = None
fxc_gamma = 2 * fxc_gamma * grids_weights
fxc_rho_full = numpy.zeros([n_dm1_batch, ngrids_full])
fxc_rho_full[:, rho_nonzero_mask] = fxc_rho
fxc_rho = None
fxc_gamma_full = numpy.zeros([n_dm1_batch, 3, ngrids_full])
fxc_gamma_full[:, :, rho_nonzero_mask] = fxc_gamma
fxc_gamma = None
g1 = 0
for split_ao, ao_mask_index, split_weights, split_coords in ni.block_loop(mol, grids, mol.nao, 1, max_memory):
split_ao = split_ao.transpose(0,2,1) # order: component, ao, grid
g0, g1 = g1, g1 + split_weights.size
split_fxc_rho = fxc_rho_full[:, g0:g1]
split_fxc_gamma = fxc_gamma_full[:, :, g0:g1]
for i_dm in range(n_dm1_batch):
# \mu \nu
V_munu = contract("ig,jg->ij", split_ao[0], split_ao[0] * split_fxc_rho[i_dm, :])
# \mu \nabla\nu + \nabla\mu \nu
nabla_fxc_dot_nabla_ao = contract("dg,dig->ig", split_fxc_gamma[i_dm, :, :], split_ao[1:4])
V_munu_gamma = contract("ig,jg->ij", split_ao[0], nabla_fxc_dot_nabla_ao)
nabla_fxc_dot_nabla_ao = None
V_munu += V_munu_gamma
V_munu += V_munu_gamma.T
V_munu_gamma = None
vmat[i_dm + i_dm1_batch, :, :] += V_munu
V_munu = None
if output_in_2d:
vmat = vmat.reshape((mol.nao, mol.nao))
return vmat
def _check_mgga_grids(grids):
mol = grids.mol
atom_grid = grids.atom_grid
if atom_grid:
if isinstance(atom_grid, (tuple, list)):
n_rad = atom_grid[0]
if n_rad < 150 and any(mol.atom_charges() > 10):
logger.warn(mol, 'MGGA Hessian is sensitive to dft grids. '
f'{atom_grid} may not be dense enough.')
else:
symbols = [mol.atom_symbol(ia) for ia in range(mol.natm)]
problematic = []
for symb in symbols:
chg = gto.charge(symb)
if symb in atom_grid:
n_rad = atom_grid[symb][0]
else:
n_rad = gen_grid._default_rad(chg, grids.level)
if n_rad < 150 and chg > 10:
problematic.append((symb, n_rad))
if problematic:
problematic = [f'{symb}: {r}' for symb, r in problematic]
logger.warn(mol, 'MGGA Hessian is sensitive to dft grids. '
f'Radial grids {",".join(problematic)} '
'may not be dense enough.')
elif grids.level < 5:
logger.warn(mol, 'MGGA Hessian is sensitive to dft grids. '
f'grids.level {grids.level} may not be dense enough.')
[docs]
class Hessian(rhf_hess.HessianBase):
'''Non-relativistic RKS hessian'''
_keys = {'grids', 'grid_response'}
def __init__(self, mf):
rhf_hess.Hessian.__init__(self, mf)
self.grids = None
self.grid_response = False
partial_hess_elec = partial_hess_elec
hess_elec = rhf_hess.hess_elec
make_h1 = make_h1
from pyscf import dft
dft.rks.RKS.Hessian = dft.rks_symm.RKS.Hessian = lib.class_as_method(Hessian)
dft.roks.ROKS.Hessian = dft.rks_symm.ROKS.Hessian = lib.invalid_method('Hessian')
